STUDIES ON HAZARD RATIO FUNCTIONS IN SURVIVAL ANALYSIS
WITH SPECIAL REFERENCE TO PROPORTIONAL
HAZARD FUNCTIONS MODELS
by
Brenda Kay Edwards
Department of Biostatistics
University of North Carolina at Chapel Hill
Insti tute of Statistics
Mime0S-~ries
June 1975
No. 1018
~'\i'!
"~.? ~".
~< .:.: --:... ,
(,.j;'.
BRENDA KAY EDWARDS.
Studies on the Hazard Ratio Functions in Survival
Analysis with Special Reference to Proportional Hazard Function Models.
(Under the direction of REGINA C. ELANDT-JOHNSON.)
The proportional hazard function (PHF) model,
where
f.l
g
(t)
PI
is the hazard function for population
TI
(t)
g
= 8f.l
(t),
= 1, Z, is
g
,
Z
used as a model for comparing survivorship functions of two (or more)
populations.
8
In Chapter II we discuss the estimation of the ratio
= ~l (t)/~Z(t)
and
~
=
8/(1 + 8)
in the PHF model.
new estimators and the maximum likelihood
(ML)
We derive three
estimator of
11.
The
properties and distributions of the estimators for small sample sizes
in two sampling schemes are presented.
We consider estimation condi-
tional on the observed numbers of individuals from
risk of dying at each of the times of death (SS1).
and
at
We also consider
estimation when the numbers of individuals in the risk sets are taken
as random variables which reflect the natural decrement occurring in the
two populations (SSII).
In SSI all four estimators are biased, the
third estimator proposed was usually found to have the smallest mean
square error (MSE) and was close to the
ML
estimator.
In SSII the
first estimator proposed was shown to be a function of the (unbiased)
Mann-\.,Thitney V-statistic, and the third estimator was usually found to
have the smallest MSE.
The PHF is a special case when the hazard ratio function (HRF)
is assumed constant for all
t, i.e. 8(t) = ~l (t)/~2(t) =
e
for all
t.
In Chapter III we investigate the more general situation in which the
HRF is a function of
For many empirical data cases we found that
t.
the HRF is not constant, except perhaps over a restricted range of
values of
t.
The use of experimental evidence to draw conclusions
about the form of the HRF may be difficult, especially for small
samples. If, however, we specify the parametric family of the failure
distributions, the HRF may be a well-behaved function of
studied analytically.
t, and is
The results presented in Chapter III suggest
that the PHF assumption may be valid only under certain conditions on
the parameter values or on the range of values of
t.
In Chapter IV we present the results of a Monte Carlo simulation
study designed to compare some "average" of the function
6(t)
= 8(t)/(1 + B(t»
mates of
"6"
and
of
at the observed times of death with the esti-
obtained under the PHF model.
1T 2
The failure distributions
are specified by a parametric model.
We also suggest
some alternatives to the PHF model and outline methods for assessing
the adequacy of the PHF model with respect to these alternatives which
are not based on any a priori knowledge of the failure dis tributions in
1T
l
and
1T
Z·
An example is given in Chapter V to illustrate the use of the
estimators of
6, proposed in Chapter II, that allow for both tied and
censored observations.
We conclude the dissertation with some sugges-
tions for future research.
TABLE OF CONTENTS
v
ACKNOWLEDGEMENTS
. . . . . . . . . . . . . . . . . vi
LIST OF TABLES
LIST OF FIGURES . . . .
I.
INTRODUCTION AND REVIEW OF LITERATURE
1.1
Introduction
1
1. 2
Definitions
4
1.3
Parametric Models
5
1.4
Nonparametric Models
5
1.4.1
Linear Approximation
8
1.4.2
Piece-wise Exponential Approximation
9
1.4.3
Step-function Approximation . . . .
10
Survival Models with Concomitant Variables
11
1.5.1
12
1.5
II.
. viii
Parametric Models with Concomitant Variables
1.6
"Quasi Parametric" Models
16
1.7
Comparison of Survival Curves
19
ESTIMATION IN THE PHF MODEL
2.1
Estimators of the Ratio
2.2
Estimators of
6
8
=
~1 (t)/~2(t)
21
and Their Properties Under Sample
Scheme I
25
6
2.2.1
Approximation to First and Second Moments of
2.2.2
Choice of Weights {a.}
27
2.2.3
Three Estimators of
28
1
6
25
iii
2.3
2.2.4
(Conditional) Maximum Likelihood Estimator of
2.2.5
Nearly Unbiased (NU) Estimators of
Estimators of
~
~
32
~
and Their Properties Under Sample
Scheme II
2.4
30
32
2.3.1
Relationship of
2.3.2
Other Estimators of
~l
to Mann-Whitney V-statistic
33
~
35
Extensions to Ties and Censoring
36
2.4.1
Ties
36
2.4.2
Censoring
40
A
2.5
Distributions of
~1
(1
1,2,3)
and
~4
and Some of
Their Properties
40
2.5.1
SSI (No Censoring)
41
2.5.2
SSII (No Censoring)
52
III. SOME ANALYTICAL AND EMPIRICAL INVESTIGATIONS OF THE HAZARD RATIO
FUNCTION (HRF)
3.1
The HRF for Some Population Mortality Data
63
3.2
The HRF for Some Empirical Data
64
3.3
The HRF for Some Parametric Models
75
3.3.1
The HRF for Some Parametric Models from the Same
Family of Distributions
..
.
77
3.3.1.1
HRF for Two Weibull Distributions
79
3.3.1.2
HRF for Two Makeham Gompertz Distributions
79
3.3.1.3
HRF for Two Linear Hazard Distributions
81
3.3.1.4
HRF for Two Lognormal Distributions
82
3.3.1.5
HRF for Two (Left) Truncated Logistic
Distributions
3.3.1.6 ·HRF for Two Gamma Distributions
83
86
iv
3.3.1.7
3.3.2
HRF for Two Burr Distributions
The HRE for Pairs of Parametric Models from
Different Families of Distributions
IV.
ESTIMATION OF
4.1
~
. . . . . 88
UNDER SOME NON-PHF MODELS
The Estimator
Delta Function
4.2
86
~
as an Estimator of an IIAverage" of the
90
~(t)
Monte Carld Simulation Study
94
4.2.1
SSI Simulation Study
95
4.2.2
Discussion of SSI Simulation Study
96
4.2.3
SSII Simulation Study. . . .
4.2.4
Discussion of SSI1 Simulation Study
4.2.5
Some Measures of the Difference Between the
Function Model and an Estimate
.101
~
102
~(t)
Based on the
107
.108
. . 109
·111
~
5.2
Which Allow for Ties
Example of ~~rtality Data Among Hourly Workers in the
. . . 112
ACKNOWLEDGEMENTS
The contributions of Professor Regina C. Elandt-Johnson as adviser
to both my academic endeavors and avocational persuits are gratefully
acknowledged and kindly remembered.
Johnson I
E~xpress
To her and to Professor Norman L.
a sincere appreciation for their guidance, encourage-
ment, availability and friendship, both jointly and separately, during
the course of this research.
I also wish to thank my other committee
members, Dr. Joan Cornoni, Dr. C. Ed
Davis, Dr. David G. Hoel and in
particular Dr. Dana E. A. Quade, for their helpful comments.
The financial support for this research and graduate study in the
Department of Biostatistics was through Training Grant GM00038-2l from
the National Institute of General Medical Sciences.
I appreciate the
opportunities which this support has afforded.
I wish to thank Dr. Norman E. Breslow for making data from the
Children's Cancer Study Group A available, the Occupational Health
Studies Group of the University of North Carolina for the data which
they provided, and J. O. Kitchen for helping me adapt to my own needs
his program for evaluating
SB
parameters.
For the skill, equanimity and good-humored cooperation which Ann
Hackney brought to the task of typing this manuscript under many changing deadlines I am most grateful.
I also appreciate Gay Hinnant for her
skillful typing of most of the tables in Chapters II and III.
For the sundry "aids" which were extended to me by family and
friends during graduate school I express a warm appreciation.
LIST OF TABLES
2.1
r'
_2 = [4 4 4 4 3 2]
2.2
.
[4 3 2 1 1 1],
.
~
7, r'
_1
SS1, Exa.ct Distribution, k
SS1, Exact Distribution, k
7, r'
_1
2.6
SS1, Sample Distribution, k
2.7
SS11, Exact Distribution, k
2.8
2l
· .
.
7, r'
_1
=
46
18, Sample Size
=
7, r
·
47
·
48
[5 4 3 3 2 2 1],
3, r
l1
21
220
=
SO
·
and
4
4
· 54
= 10, r 1l = 5, r 2l = 5
4,
ll
.
r
2l
= 5 and
·
55
Chi-square Goodness of Fit Test Statistics, Exact Distributions,
SSII
2.10
.
.
SS11, Exact Distribution, k = 9, r
k
2.9
4, r
45
.
r"
_2 = [5 5 5 4 4 3 3]
=
· · . ·
[4 4 4 4 4 3 2],
.
SS1, Exact Distribution, k
= 8, r l1
44
8, r'
_1 = [4 3 3 3 3 2 2 1] ,
SS1, Exact Distribution, k
k
·
·
·
. ·
r'
_2 = [5 5 4 3 2 2 1 1]
2.5
· .
· .
r'
_2 = [5 4 3 2 1 1 1]
2.4
.
[4 4 4 3 2 2 1],
. .
r'
_2 = [4 3 2 2 2 1 1]
2.3
6, r'
_1
SS1, Exact Distribution, k
·
SS11, Sample Distribution,
Sample Size
= 186
k
=
20, r
11
=
10, r
2l
=
. . • 57
10,
• . • . 59
vii
2.11
I~ D
Kolmogorov-Smirnov One Sample Test Statistics
Sample Distribution, Sample Size
SSll,
n
= 186 . . . .
61
.···
.·
. . ·
3.1
Example 1, Sample Estimates of HRF
3.2
Example 2, Sample Estimates of HRF
3.3
Example 3, Sample Estimates of HRF's
3.4
Some Parametric Families of Distributions
3.5
Summary of the Behavior of the HRF for Some Parametric Hodels
..
.
··
· · ·
.
· ··
· · · .
·
·
·
.
·
· · · .
69
71
76
from the Same Family of Distributions
3.6
67
78
Parameter Values of the HRF for Pairs of Lognormal
Dis tributions
84
3.7
Parameter Values of the HRF for Pairs of Gamma Distributions
87
4.1
Summary of 200 Sample Estimates of "6" Under Some Specified
97
Parametric Models, SSI . . . . • .
4.2
Summary of 200. Sample Estimates of "6" Under Some Specified
Parametric Models, SS11
5.1
· 103
Summary of Data on Six Populations of Workers in a Major
Tire Manufacturing Company, Akron, Ohio, 1964-1973
5.2
Estimates of
in a
~fujor
~
118
for Three Comparisons of Populations of Workers
Tire Manufacturing Company, Akron, Ohio, 1964-1973. 119
LIST OF FIGURES
3.1
Males and Females, United States 1966 Population . . . .
65
3.2
Nonwhite and White Males, United States 1966 Population
65
3.3
United
Sta~es
(Males) and England and Wales (Males), 1967
Population . . . . . . . . . .
66
3.4
United States (Males) and West German (Males), 1967 Population
66
3.5
Example 1, Plot of Estimated HRF
68
3.6
Example 2, Plot of Estimated HRF
70
3.7
Example 3, Plots of Estimated HRF's
73
3.8
HRF for Pairs of Weibull Distributions
80
3.9
HRF for Two Nakeham Gompertz Distributions
81
3.10
HRF for Two Linear Hazard Distributions
82
3.11
HRF for Two Lognormal Distributions
85
3.12
HRF for Two Truncated Logistic Distributions
85
3.13
HRF for Two Gamma Distributions
87
3.14
HRF for Two Burr Distributions
4.1
Histograms of 200 Sample Estimates of
.
88
f.,
When
Mt)
is
Specified by Two Weibull Distributions, SSI
4.2
Histograms of 200 Sample Es'timates of
!'I
When
• . . . 98
[\(t)
is
Specified by Two Gompertz Distributions, S81 .
4.3
Histograms of 200 Sample Estimates of
f.,
When
· . . . 99
f.,(t)
is
Specified by Two Weibull Distributions, S81
4.4
Histograms of 200 Sample Estimates of
f.,
When
Specified by Two Weibull Distributions, SSII .
· . . . 100
f.,(t)
is
· . . . 104
ix
4.5
Histograms of 200 Sample Estimates of
6
When
6(t)
is
. . . . 105
Specified by Two Gompertz Distributions, SS11
4.6
Histograms of 200 Sample Estimates of
6
When
Specified by Two Weibull Distributions, SSII
6(t)
is
. . . . 106
CHAPTER I
INTRODUCTION AND REVIEW OF LITERATURE
1.1.
Introduction.
The random variable of interest, T, say, in many epidemiological
studies is the length of time elapsing until an individual experiences
a well-defined event, often termed "death" or "failure".
The distri-
bution function of this random variable, i.e. the cumulative failure time
distribution, its complement (often called the survival function), and
the instantaneous rate of failure (generally termed the hazard function),
are of both practical and theoretical importance.
The survival function
and the hazard function are frequently used to describe and summarize
the results of follow-up studies in epidemiological investigations.
These longitudinal or prospective studies (cohort studies) are basic
designs in biomedical research.
Implicit in the design is the definition
of the population (cohort) to be followed for a specified period of
time.
Also, the outcome event which may occur in that population during
the follow-up period must be well-defined.
Incidence studies, clinical
trials and intervention trials are important areas of application of
follow-up studies.
Statistical methods for estimating the functions that describe
survivorship depend on the manner in which measurements on tile random
variable
T
are obtained.
Problems in the
briefly summarized into three groups.
(i)
censored observations,
analysi~ ~f
the data are
These are associated with
z
and
(ii)
competing risks,
(iii)
concomitant variables (or covariables).
Before describing these problems it is first necessary to distinguish between two periods of time:
period.
the study period and the follow-up
The study period is the interval of time measured from the
beginning until the termination of the study.
It may be expressed as
the actual calendar time or as the number' of units of time, usually in
months or years, of the study period.
The follow-up period is the length of time each individual is
observed during the study period; the time he enters the study period
is the beginning of his follow-up period, i.e.
to
=
O.
The survival
information contributed by each individual during his follow-up period
is used to estimate the survival function.
The follow-up period for
each individual is of variable length and is limited by the closing
date of the study.
In this case the survival time observation is
(right) censored.
Intervening events, i.e. deaths from other causes (competing
risks), may also rer,nove the individual from observation before the
study is completed or before the event of interest has occurred.
censoring and competing risks
produce
Thus,
partial information on survi-
vorship for each individual.
On the other hand, incorporation of measurements on variables in
addition to survival time poses different problems.
It is usually
assumed that survival time is some function of this additional information, or a function of certain concomitant variables.
Let
t
l
< t
z<
••• < t
k
represent the observed (ordered) times
of failure during the follow-up period.
Without loss of generality we
3
will refer to them as the times of "death".
K
i
(more precisely
as the set of individuals alive and at risk
i
of dying at
(i
R )
t
We may define the risk set
t
i
= l, •.. ,k).
, with
r
i
denoting the number of individuals in
In general, R
i
in common, e.g.
R +
i l
and
R.
1
need not contain any members
when the distinct times of death represent data from
different independent experiments.
Estimation of the survivorship
functions could be conditional on the underlying risk sets
Such
{R.} .
1
a situation occurs at the conclusion of a study when estimation is conditional on
t~e
observed risk set for each time of death.
We will refer
to estimation of the survivorship functions which is conditional on the
risk sets as Sample Scheme! (SSI).
In most situations, however, such as follow-up studies, the set
represents the survivors out of
in
[ti,t + ).
i l
R.
1
who were not censored or lost
In these particular circumstances
contain members in common.
R
i
and
R+
i l
do
Estimation of the survivorship where the
risk sets are considered as random variables will be called Sample
Scheme II (SSII).
The remaining parts of this chapter will give a brief review of
literature on methods available for estimating and comparing survival
functions.
Special models based on the assumption of a proportional relationship between two hazard functions (the PHF model) which are frequently
utilized in the estimation and comparison of survival functions are
considered in Chapter II.
Three new estimators of the parameter in the
PHF model, their distributions, and their properties for Sample Scheme I
and Sample Scheme II are given.
4
A discussion of some empirical evidence and theoretical conditions which illustrate the inappropriateness of the PIIF model in
certain circumstances is presented'in Chapter III.
In Chapter IV some consequences of estimating the parameter in the
PHF model when an alternate model is valid are given.
"tests" of the PHF model are suggested.
Two
indirec t
Chapter V contains an example
of the application of the methods of analysis of survival data investigated in this paper to ten years of follow-up data from a population of
workers in the rubber industry.
1.2.
Definitions.
We briefly review the basic definitions useful in survivorship
analysis.
Let
T
be a positive continuous random variable representing
the lifetime of an individual.
Cumulative Distribution Function
F(t)
F(O)
Pr{T
= 0;
~
t}
=
(cdf)
Pr{individual dies at or before time
F(oo) = 1.
t}.
(1.2.1)
Survival Function
pet)
P(O)
Pr{T > t} = Pr{individual survives longer than time
= 1; P(oo) = O.
t}.
(1.2.2)
Hazard Function
Pr{t < T < t + dtlT > t}
~(t)
lim--------:-~-----
d t-+O
dt
d
<ft9-n[P(t)],
where
f(t)
d
--F(t)
dt
d
dt:P(t)
ICt)
P (t)
,
(1.2.3)
(1.2.4)
is the probability density function
5
(pdf) of tile distribution function of times of death.
The hazard
function is also called the force of mortality and is, in fact, the
instantaneous death rate.
From (1.2.4) we have the relations
t
pet)
exp[-! ~(T)dT].
(1.2.5)
~(t)exp[-! ~(T)dT].
(1.2.6)
o
f(t)
o
It is only necessary to specify the form of anyone of the functions to
obtain the form of the other two.
1.3.
Parametric Models.
Parametric models (Buckland, 1964; Johnson and Kotz, 1970)
frequently applicable to the descrjption of lifetime distributions will
be discussed in Chapter III.
hazard function, or
Poisson Process.
deaths
A standard reference point is a constant
~(t) = ~
for all
t; it is associated with the
A constant hazard function is equivalent to assuming
occur at random.
The distribution of times of death is
described by a (negative) exponential distribution.
Monotone hazard
functions correspond to always positive or negative wearing mechanisms.
The linear hazard function model, Weibull, Makeham Gompertz, gamma, Burr,
and (truncated) logistic families of distributions all have monotone
(increasing or decreasing) hazard functions.
For some distributions,
e.g. the lognormal distribution, the hazard function is not monotone it is increasing for some values of the variable and decreasing for
others.
1.4.
Nonparametric
Mode~.
The classical approach to estimating survivorship functions has
6
been through the use of life table methods developed by actuaries.
Life table functions which are estimated from
information contri-
buted by the cohort of individuals in a follow'-up study are defined
below.
w non-overlapping intervals,
Divide the follow-up time into
[t ,t + ), each of width
i i 1
{l }
i
hi (i = O, ... ,w - 1), to = O.
is the nonincreasing sequence of values representing the
expected numbers (or propcirtions) of individuals surviving
to at least
(when
1
=
0
t
out of
i
1
0
individuals alive at
to
1).
More precisely, this should be
1
t.
, but for convenience we will simply
1
write the subscript
t.
i
for all the life table functions that depend on
unless ambiguity arises, (e.g.
1
1
t.
1
{t.}
1
= 1.).
1
Notice that the set
now represents interval endpoints of failure rather than observed
failure times.
p.
1
Pr{surviving to at
pdT> t.}
1
~east
t. }
1
t.
exp[-J
o
1
lJ(t)dt]
t
-J
t
2
t.
lJ(t)dt - ... -
J1
lJ(t)dt]
t.1- 1
l
i-I
n
j=O
where
p.
J
(1.4.1)
7
exp[-J
Pi
t i +l
~(T)dT]
(1.4.2)
t.
1
= PdT>
ti+llT > t i }
P +
p.
£i+l
l
-i -
= -£.1
that is, p.
1
(1.4.3)
1
is the conditional probability of surviving at least
additional years, given alive at
q.
1
=
1 - p.
1
=
h.
1
Its complement
t ..
1
1 - exp[-J
t i
+l
(1.4.4)
W(T)dT],
t.
1
is the conditional probability of dying in the interval
given alive at
t ..
1
Expr~ssion
[ti,t + ),
i l
(1.4.1) illustrates that the survival
function can be represented as the product of the conditional probabilities of surviving successive intervals of time.
d.
1
t
Jt. i
+l
W(T)£ dT
=
T
~.
(1.4.5)
1
1
where
d.
1
is the expected number of deaths occurring in the interval
[ti,t i + l )·
/i+l p(TH dT
T
t.
m.
1
d.
1
Jt i + l
t.
~
1
Jt i + l
dT
T
where
m.
1
1
is the central death rate.
individuals alive at
t ..
1
per person per unit of time.
dT
T
1
The denominator of the function
is the expected number of years lived between
~.
~
t.
1
(1.4.6)
Thus
m.
1
and
t.
1
t i +l
by the
is the expected number of deaths
Precise actuarial notation would be
Pt.
1
and
m
t.
1
For convenience we will use
Pi
and
m.•
1
8
Life table methods are primarily counting techniques wllich estimate the denominator of (1.4.3) or (1.4.6) using
on survival.
partial information
These methods have been used in their simpler forms by
epidemiologists since 1910 (Elderton and Perry, 1910, 1913; Weinberg,
1913) .
Two approximations to the survival function which have been used
with grouped data, and a method used with ungrouped data will be discussed.
Their application to the analysis of follow-up data from epidemio-
logical studies is cited.
1.4.1.
Linear Approximation.
In analyzing a set af follow-up data grouped into intervals it
may be reasonable to make the following assumptions.
(i)
Times of death are uniformly distributed over each
interval.
(ii)
Each person leaving (or entering) the experience contributes,on the average, to one-half of the interval length
of exposure.
(iii)
Events in different intervals are independent.
A uniform distribution of deaths in each interval (i) is equivalent to assuming the survival function is linear in the interval.
may be necessary to group the data into intervals of
unequal
in which this assumption holds (Merrell and Shulman, 1955).
It
lengths
For
example, infant mortality or post-operative mortality is approximately
linear over small intervals of time, whereas later follow-up mortality
is effectively linear over much larger intervcus of time.
In 1933 Frost used the actuarial method and assumptions (1)-(1ii)
to obtain an estimator of
m , the age-specific annual death rate,for a
i
9
study population exposed to pulmonary tuberculosis.
The denominator of
the estimator was defined in terms of person-years. or "risk time units"
(one person exposed to the risk of dying for one year).
In the 1950's
the same methods were applied to estimate the conditional probability
of dying, q .• using partial information (Berkson and Gage. 1950; Cutler
1
and Ederer. 1958).
Actuarial estimators of
Number initially exposed to risk in
[ti.t +l ) •
i
and their corresponding large sample variances, have also been obtained.
The earliest· known derivation of the approximate standard error of the
estimated survival function was given in 1926 by Greenwood (see also
Irwin, 1949; Cutler and Ederer, 1958; Kaplan and Meier, 1958).
For a
review of actuarial methods used to estimate survivorship functions
see Gehan (1969).
1.4.2.
Piece-wise Exponential Approximation.
We now consider the situation where assumption (i) is replaced by
the assumption that the hazard-function
the interval
[ti,t + ), (i
i l
= O, ...• w -
~(t)
1).
is a constant
~i
over
The number of deaths in an
interval has a binomial distribution with the probability of death given
by (1.4.4).
to
=1 -
q.
1
Using this assumption the probability of death simplifies
exp(-~.h.).
1
1
A maximum likelihood (ML) estimator of
~1'
based on this assumption can be derived in terms of the MLE of
p.
1
=1
- q.
1
(Gehan, 1969).
other "MLE's" of
(i)
~i
It is worth remarking at this point that
can also be derived under other assumptions, if
and (iii) hold (see Elveback (1958)).
Where there are no losses Chiang (1968) used the assumption of a
10
constant hazard function over each interval of folJow-up time to
e~timate
the survival function.
He also divided tilt' risk sets
into two independent populations:
those
II
due "
{R.}
1.
•
for censoring
in tht'
interval and those not "due" for censoring in the interval.
When the
number of deaths among those due to be censored is zero, his estimator
of the conditional probability of survival,
actuarial estimator.
Pi'
is equivalent to the
Also, when the number due to be censored and/or
the hazard rate function in each interval is small,it has been shown that
there is no practical difference between actuarial and Chiang's estimation of
(Kuzma, (1967)).
When the hazard function
jJ.
is small there is essentially no
1.
difference between the estimators of the survival function obtained by
linear approximation and by piece-wise exponential approximation.
is obvious, for when
1J
i
This
is small the conditional probability of death
in each interval is approximately uniform, i.e.
q. = 1 - exp[-jJ.h.l
1
1.
1.
- 1-1. h. .
1
1
(1.4.7)
In this case the survival function is approximately linear over the
interval.
1.4.3.
Step-function Approximation.
If the times of each single death are available the survival
function can also be approximated by the "empirical" distribution, i.e.
a step-function with jumps at the times of death.
Since the estimated
conditional probability of survival is equal to one over intervals in
which no deaths occur, it is only necessary to consider the intervals
determined by the times of death.
Kaplan and Meier (1958)" called this
11
the product-limit (PL) estimator of the survival function.
Turnbull
(1973) extended the PL estimator to doubly censored data, i.e. data
that is both left and right cens?red.
It should be noted that the three methods - life table with
linear approximation, life table with exponential approximation, and the
PL step-function approximation, have been developed for analysis of
survival data at the close of a study, conditionally on the risk sets
{R.}.
1
1.5.
These methods are in the frame of Sample Scheme I.
Survival Models with Concomitant Variables.
It is usually reasonable to assume that the probability of sur-
viva1 in the presence of a morbid or lethal condition is a function of
one or more variables in addition to time or age.
termed explanatory or concomitant variables.
These have been
The experimental design
of a follow-up study usually incorporates some such variables as control
factors.
However, the analysis of survival data may need to use infor-
mation contributed by these variables, i.e.
(i)
randomization procedures may fail to achieve a balance
of some factors among populations,
(ii)
it may be impractical to control for a large number of
factors, or
(iii)
the importance of the variables appears after the study
design is implemented.
The use of models of survival which include concomitant variables
(also sometimes called regression models)
four
or covariance analysis has
main purposes.
(i)
To inquire whether the explanatory variables are significant in the model (McGilchrist, 1974; Greenberg
e~
al.,
12
1974).
(ii)
To correct for lack of comparability with respect to the
concomitant variables among groups whose survival is being
compared (Feigl and Zelen, 1965; Glasser, 1967).
(iii)
To select optimum treatment assignments for patients in a
clinical trial (Aitchison, 1970; Byar, 1974).
(iv)
To reduce the variance in the survival time (or response
variable) (Armitage and Gehan, 1974).
1.5.1.
Parametric Models with Concomitant Variables.
Adjustments of observed survival times to account for concomitant
variables have primarily been developed by modeling
the functional
dependence of the hazard function (and therefore of the survival distribution) on the covariables.
exponential model.
The simplest formulations have been for the
It has often been shown that this model describes
survival following diagnosis of various types of cancer (Myers et al.,
1966).
Other models, notably the multivariate logistic, the Weibull,
the log gamma and the Gompertz have been considered.
The exponential
is a special case of all these models except the logistic.
In 1965 Feigl and Zelen suggested an exponential model to describe
survival of leukemia patients.
Let
z
be the observed value of the
random variable corresponding to white blood cell (WBC) count, and
60
be the parameter for effect due to membership in a specific population.
The expected survival time of each individual in that population is then
assumed to be a linear function of the concomitant variable
E[T]
jJ
-1
z, i.e.
(1.5.1)
13
or
~
Inclusion of the parameter
8
>
o.
(1.5.2)
allows the survival function to vary
1
among individuals according to the value of each person's WBe count
measured at some point, usually
to'
The likelihood function to be
maximized is the product of exponential density functions where the
parameter
in (1.5.2).
~
= g(Z;8 0 ,8 l )
is a function of the covariable and parameters
This model was extended by Zippen and Armitage (1966) to
include censored observations.
nential model where
= 80
~
Byar et ale
(1974) utilized the expo-
+ BIZ.
Concurrently Glasser (1967) suggested an exponential model to
adjust censored survival data following surgery for lung cancer
in two
populations which differed with respect to their age distributions.
Let
z
be the deviation of an individual's age from the average age of
the total sample populations, and
~O
be the hazard function for an
individual whose age is equal to the average age of the total populations.
The mean survival of each individual is then assumed to be
defined by
E[T]
~
-1
,
(1.5.3)
where
~
= Wo exp(Sz),
This type of model is usually termed a log-linear model.
(1.5.4)
Maximum likeli-
hood estimation of the parameters in this model, generalized to more than
one covariable, has recently been given (Breslow, 1974a).
A log-linear
model for each of a number of subintervals of the follow-up period has
been suggested (Holford, 1972).
This results in a piece-wise expo-
nentia1 model for the overall survival function of each individual.
The multivariate logistic function has been proposed (Truett
~
aI., 1967; Walker and Duncan, 1967) as a model for fitting a dichot-
omous response variable (such as death or survival) to a number of
independent covariab1es.
In this model it is assumed that the uncon-
ditional probability of dying depends on
m covariab1es
z
according
to the relation
Q(z)
[1 + exp(-S'z)]
-1
exp(S'z)/[l + exp(B'z)],
where
(1.5.5)
B is the corresponding vector of parameters to be estimated.
A
generalization of the model to fit a categorical response variable witll
more than two outcomes has also been proposed (Mantel, 1966a).
This
model has the property that the logarithm of the ratio of the probability of dying to that of surviving is a linear function of the covariables, I.e.
Q(z)
log [-1
Q(z)]
m
I
B,z.
(1.5.6)
j=l J J
This mode1,describes the response over varying units of time.
It is
possible to include the actual length of follow-up time in the model as
one of the covariables (see Biometric Society Committee's Report on
Trials of Hypoglycemic Agents, 1975);however, tIl is implies that response
time is a covariab1e.
Extensions of the multivariate logistic risk function to take
into account individual response time information have also been
15
suggested (Myers
~~ a~.,
1973).
intervals of equal length
stant force of mortality
h.
~
Let the follow-up period be divided into
Assume each individual has the same con-
per unit length of such intervals.
probability of survival over any unit is
p
=
exp(-~h).
The
The probability
of surviving each of the units preceding the unit in which an individuals's first death occurs is given by a geometric distribution with
parameter
q
=
1 - p
=
exp(-~h).
1 -
Further, it is assumed that the
log odds of survival to death for each interval satisfies the "linear
logistic
functio~'
. 10g(1 -h
p
where
z
is a
m x 1
vector of parameters.
Ph
)
=
Y6.z.
(1.5.7)
,
j=l J J
S
vector of covariables and
the corresponding
Rearrangement of (1.5.7) gives the hazard function
of the exponential distribution in terms of the covariables, i.e.
1
h
m
log[l + exp( I 6.z.)].
(1.5.8)
j=l J J
This model has been called the logistic-exponential regression model.
It can be shown that the limiting case when
linear model; when
h
h
is zero is a log-
is infinite the hazard function is
linear in the regressor variables (i.e. the exponential model where
~ =
m
I6.z.).
A suggestion for extending the model to permit individual
j=l J J
variation in the hazard function was also given in this paper.
Other parametric models have been proposed which include concomitant variables in various functional relations to the survivorship
functions.
Theoretical results for fitting a normal distribution by
weighted linear regression to the logarithm of a Gompertz hazard
function have been described (Prentice and El Shaarawi, 1973).
16
For small class intervals and short follow-up periods age-specific
death rates can be considered as estimating the hazard function.
It
was suggested that covariables related to age class could be included in
an exponentially multiplicative manner in this model.
has also suggested a Weibull model.
Prentice (1973)
This is one of several parametric
forms which allows the hazard function to depend on time and on a set
of covariables which are independent of time.
Inclusion of covariables
in a generalized log-gamma model were presented elsewhere (Prentice,
1974).
Estimation and tests of hypotheses become rather complicated in
these last two models.
Choice among models, particularly regarding the manner in which
concomitant variables affect the survival function as expressed through
the hazard function, has only recently received much attention.
Fisher
and Kanarek (1974) have proposed a metric to measure the "distance"
between the survival curves of different models which are functions of
only one covariable.
Their primary concern is
the?j~~
of the "di ffer-
ence" between models, not methods of discriminating between models.
As
yet they have published only preliminary results of their investigation.
The choice between models is not only a question of which model fits the
data best; it is also a question of which set of assumptions is the most
reasonable representation of the physical process heing studied.
1.6.
~'Qua~_~~~_~_a2J1..e_s_ri~'_ Hodels.
In other recent work the comparison of survival functions has
been approached using essentially "nonparametric" methods, but assuming
the survival functions are in a specific (parametric) relationship to
each other.
In particular, the
model has been assumed.
proportiona~
hazard
..functiC:'..!.~
For two populations, this specifies
(PHF)
17
e
where
~
g
(t)
> 0,
(1.6.1)
is the hazard function of population
g
1,2
=
at time
t.
A "generalized proportional" relationship among hazard functions
was assumed by Cox (1972),
t
where
~'=
(z z2 ... zm)
l
B
for an individual,
~o(t)
B = 0.
is a
1
x
(1.6.2)
0,
m vector of concomitant variables
is the corresponding vector of parameters, and
is the (unknown) hazard function for the condition
The vector
z
z
=
If
B
equations for the ML estimators of
obtain~d
or
is a function of time
z
it must not involve either of the two hazard functions
Cox
°
may be zero-one indicator variables; in this
model they may also be functions of time.
(z(t»
~
~(t;z)
based on a
conditional likelihood function conditioned on the actual times of death.
Consider the probability
Pr{individual (i) dies in
(£)ER.
[ti,t
+ dt i ) Ian individual.
i
[to ,to + dt.)}
dies in
1 1 1
1
TT
[~.(t.)dt.][
(1 - Pn(t.)dt.)]
1 1
1 £E(R.-i)
~
1
1
1"
d
1
t~:a IR {[PJ. ( t . ) d t . ][ £ (R1T_') (1
1
.
1
JE i
1
E
i
J
- Pn( t
~
l' )
dt
7
iJJf
P. (t.)
1
1
L P.(t.)
. R J
JE i
1
where the instantaneous probability of individual (i) dying in
(1.6.3)
18
[ti,t. + dt.), conditional on being alive at
l.
~.(t.)dt.,
l.
l.
t., is very nearly
l.
l.
i
=
1, ... ,k.
1
Elimination of the nuisance parameters in
makes the likelihood depend only on
8
and
In practice "ties" may result from measurement errors or slight
grouping of the data.
The continuous model (1.6.2) becomes complicated
to apply, regardless of how one approaches estimating
B.
COX (1970)
himself defines a different model, called the "logistic" model, when
the information is recorded with more than one death occurring at
t ..
1
This logistic model specifies the log odds of the instantaneous probabilities of death relative to survival in the interval
[t,t + dt)
as
~(t;z)dt
1 -
(1.6.4)
~(t;z)dt
The results obtained for estimating
8
using the conditional
likelihood are analogous to those appropriate for sampling without
replacement from the fixed finite populations of the risk sets
in which the selection of an individual from
R.
1
(fixed) death time is an independent experiment.
{R.} ,
1
at each observed
As Cox points out,
however, the risk set depends on what happened at previous time points
and on any intervening censoring.
Kalbfleisch and Prentice (1973) considered the same functional
relationship for the hazard function as that proposed by Cox (1.6.2).
Marginal likelihood estimators of
B were obtained which depend only
on the information contained in the rank statistics of ordered times
of death.
This approach, however, does not allow the covariables to be
functions of time.
For both censored and uncensored data, expressions
for the estimators of
8
are equivalent to those given by Cox's con-
ditional likelihood when the covariables in Cox's model are those
19
permitted by Kalbfleisch and Prentice's underlying assumption, i.e.
covariables which a"re not functions of time.
However, generalization
of the continuous model (1.6.2) to include ties results in estimators of
8 which are not equivalent to those in Cox's logistic model.
When it is unreasonable to assume a covariable (or factor) has a
multiplicat~ve
effect on the hazard function, Kalbfleisch (1974b) pro-
poses that one assumes Cox's model is appropriate for each of the
levels (or strata) of that covariable.
The likelihood function is thus
the product of the likelihood for each of the levels of the factor.
For
data including censored observations and covariables which are not functions of time, the rank order statistics of the times of death are also
sufficient for the ML estimators of the parameters in this model.
1.7.
Comparison of Survival Curves.
Comparison of survival curves is a very broad topic.
In general
comparisons have been made either in terms of tests on parameters in
various parametric models or tests based on rank statistics.
the publications on this topic will be briefly mentioned.
A few of
Results which
are relevant to the discussion in Chapter IV will be given in more detail
at that point.
"Consider the model (1.6.2) where
variable
equal to
1
z
is a single indicator
if an individual is from
if an individual is from
and equal to
o
Then
(1.7.1)
~(t;z)
This is a special case of the PHF model given by (1.6.1),
and methods which are concerned with estimating
e
or
B,
This model,
have the
advantage that data from each population contribute information to the
20
estimate of the hazard function in the other population.
A dis-
advantage lies in the restrictions on the estimates of the parameters
in the model, i.e. they must be proportional.
It should he noted that
the PHF model is equivalent to assuming the relationship of the two
survival functions is
(1.7.2)
In hypothesis testing, for
e
i 1, this is a form of the well-known
Lehmann alternative.
Numerous standard nonparametric tests for comparison of survival
(distribution) functions have been extended t6 account fot both censored
data and more than two groups (Halperin, 1960; Gehan, 1965a, 1965b;
Mantel, 1966b, 1967; Efron, 1967; Breslow, 1970; Peto and Peto, 1972;
Thomas, 1974).
An evaluation of the power of several parametric and
nonparametric two-sample tests for small samples from exponential and
Weibull distributions, with and without censoring, has recently been
presented (Lee
~
al., 1974).
The efficiency of Cox's model (1.6.2)
for the special case of the PHF model has been considered (Kalbfleisch,
1974a) for special cases:
model and (2)
(1)
assuming an underlying exponential
assuming an underlying Weibull model with common shape
parameter, for the form of
~O(t).
McRae and Thomas (1972) presented
both point estimation and tests of fit for two populations with the PHF
model and arbitrarily right censored data.
Breslow (l974b) discusses
the PHF model in terms of its relationship to the analysis of survival
data using contingency tables.
The literature cited reviews briefly
the main methods currently available for comparison of survival functions
using nonparametric tests.
CHAPTER II
ESTIMATION IN THE
PHF
MODEL
As indicated in Chapter I, the proportional hazard function (PHF)
~l (t)
model,
tion
g (g
=
e~2(t),
= 1,2)
~
where
at time
g
is the hazard function for popula-
(t)
t, has received much attention as a model in
the analysis of survival data.
It has been used as a model to estimate
and to compare survivorship functions, and to descr,ibe the relationship
of non-time dependent covariables to survivorship functions.
In this
chapter,three new estimato'rs and the MLE for a 'function of the parameter
e
are given.
The properties and distributions of these estimators
under Sample Scheme I and Sample Scheme II are also presented.
2.1.
e=
Estimators of the Ratio
tJ (t)/~2(t).
l
Consider the case of two populations denoted by
R (t)
g
r (t)
g
denote the risk set of
r.
g
(g) who survive until at least time
t,
denote the number of individuals in
R (t),
t
g
g
t
For any model
exp(-!
o
= 1,2.
g
1,2.
ilnd simply write
Recall, by definition (1.2.5)
P (t)
g
Let
TI, or those members of population
g
When there is no confusion we will drop the
and
and
TIl
g '= 1,2.
R
g
22
p
Pr{member of
o
where
P . (t)
1T
L {[
= {'
~ERI
TIl
dies firstlRl
survives to at least time
t,
(2.1.2)
mER 2
is the probability that the
gJ
(2.1.1)
R }
Z
1T P2m (t)]f Q(t)dt},
Pln(t)
nE(Rl-~)
and
g = 1,2.
j th individual in
TI
g
This simplifies, if all
members of anyone population have the same survival function, to
(2.1.3)
Using the relationships defined in (1.2.3) and (1.2.4) we obtain alternate expressions for (2.1.3)
00
J [Pl(t)]
p = rl
J [P1(t)]
o
l
r
1[P (t)] 2
2
o
00
= r
r
r
~l(t)dt
(2.1.4)
r
1[P (t)] 2[_ - } - ~--P (t)]
2
P1 (t) dt 1
(2.1.5)
And so for the PHF model substituting P (t)=[P (t)]8,i.e.,e Quation (1.7.2),
2
l
t
00
r1
J
o
r]_
J -8(r 1 8
o
[exp(-J
0
00
~1(T)dT)(rl + r2/8)]~1(t)dt
1 d
+ r )- [-- exp(-!
dt
2
t
~l(T)dT)(rl + r /B)]dt
2
0
t
-(r18)(r18 + r 2 )-1[exp(-J
o
~1(T)dT)(rl
00
+ r2/8)
1
]
0
r18
= r18 + r '
2
(2.1.6)
since
00
exp(-! ~ (T)dT)
o g
P
g
(00)
0,
0
exp(-J
a
~g(T)dT)
P (0)
g
=
1,
g
1,2.
23
Let
denote the observed distinct (ordered) time of the i
independent death in the two populations, i
R.
gl
denote the risk set of
(g)
r.
gl
=
th
l, .•. ,k,
TI , or those members of population
g
who survive up to time
t., g
1
= 1,2,
denote the number of individuals in
i
R., g
gl
= l, ... ,k.
=
1,2,
i = l , ... ,k.
Then by a similar argument to that leading to (2.1.6), under the PHF
model
Pi
=
Pr{member of
dies first!R li ,R 2i }
TIl
rlie
(Z.1. 7)
A natural unbiased estimator of
'"
Pi =
{:
'"
The estimator
Pi
i f a- member of
TIl
dies first, given r 1i , r Zi '
i f a member of
TI Z
dies first, given r 1i , r Zi '
has a Bernoulli distribution with
E[p.]=p
1
Var[p.]
1
o
is
= p.(l
1
rlie + r Zi
i
- p.)
erlir Zi
1
if the times of death are independent.
24
From equation (2.1.7)
At each of the fa ilure times
parameter
Pi
e
(2.].}\)
l, ... ,k.
i
is defined by the corresponding
and the numbers at risk
r
li
and
r
2i
; it is the
weighted odds ratio of the probability an individual from population
to the probability an individual from popula-
(1) dies first
A
Substitution of the estimator
Pi
in
(Z.1.8) would in fact give values of zero or infinity for
e.
tion (2) dies first.
e
problem can be avoided by combining the estimators of
This
at each of
the failure points in the following way.
Expression (2.1.8) may be written as
(r 11
rZ')p·
· e +1
1
Let
{a.}
1
denote
= r 1 1,e.
(2.1.9)
.
k
be a set of arbitrary weights, usually positive, and let
L
I
The sum of weighted values of (2.1.9) gives
i=l
k
L a.(r
1 1· e +
1
i=l
rZ·)p·
1
1
=
(2.1.10)
from which
Ia.rz·p·
1
1 1
8
(2.1.11)
La.rl·(l
- p.)
111
Substituting
Pi
in (2.1.11) gives an estimator of
8
Ia,r
11
2 ·p·
1
2
P.)
8
(2.1.12)
a . r ·(1 111
1
(2.1.13)
2S
at which
is the sum over all time points among
where
an individual from
TI
dies,
g
=
g
1,2.
To avoid an estimator with possibly infinite moments, which would
occur in the case of all deaths in
(giving
e
preceding all deaths in
an arbitrary monotonic increasing function of
00),
e
can
be defined by
e
= --'---
!J.
Notice that
!J.
+
I
(2.1.14)
e
is the probability that, given one person chosen at
random from
and one from
is in the bounded interval
We shall estimate
6
!J.
TI
2
dies first.
, the person from
[0,1].
by statistics of the form
e
=
1
+
e
It
La.r
·p· 1
+ La.rl·(l
1
11
21
(2.1.15 )
P.)
1
(2.1.16)
The choice of the set of weights {a.} will be considered in Section 2.2.2.
1
2.2.
Estimators of
6
and Their Properties
Yn~~ ~ampl~
Recall that Sample Scheme I (SSI) denotes estimation
e)
Scheme
of
I.
6 (or
which is conditional on the observed risk sets at every time of
death, i.e.,the numbers at risk are fixed values.
estimation of
2.2.1.
!J.
We will now consider
under SSI.
Approximation to First and Second Moments of
6.
Under certain conditions, in SSI we can get a good approximation
to the expected value and variance of
!J.
by using statistical
26
differentials.
Let
Y = \a.rl.(l
- P.),
L 1
1
1
Z
=X+
= LairZiP i + Lairli(l - Pi)'
X
~
b.
=Z '
~
= E[X]
n=
2
Y
E[Y]
\a,r 1·p·,
2 1
L 1
Ia.r1·(1
- p.),
111
.
2 2
\a,rZ·p,(l
- p.),
L 1
111
Ox = Var[X]
O~ = Var[Z]
°XZ
= Cov[X,Z]
e -- rn '
b. =
e
1 + 8
Ia:r
, - r1,)p.(1 - p.),
Z ·(r
1
121 1 1
1
=
~
~
+ n '
and
Then
(2.2,1)
27
(using
~
= en),
(2.2.2)
Bias[ll]
- ll{(~
(2.2.3)
2.2.2.
Choice of Weights
{a. }.
1
It is reasonable to choose the set of weights
optimal condition is achieved.
One way of choosing
minimize the (approximate) variance of
respect to
a.1
for fixed
II
{a.}
1
so that some
{ail is to try to
as given by (2.2.2) with
Although
is not necessarily
unbiased (2.2.3), minimizing the (approximate) mean square error of
II
would yield °no better results than minimizing the (approximate)
variance since
- 2
[Bias(ll)]
contains terms of order four.
The approxi-
mation for the variance ignores terms of order greater than two.
Let
28
Thus
Var[6]
d
-[Var(6)] - {
dct
i
(~
nE;
+
n)
4}{2 ct .r 1 .r 2 .} +
~
~
~
{--!l~4}{r2'p.}
(~ + n)
~
1
i I , ... , k.
Set the above expression equal to zero and solve for
ct .•
1
4
ct; -
•
{~2~-}{
[ . 4n; --5] [r 1 · (1
n~ r r
(~+ n)
~
1i 2i
- [-lL+ 4
(~
-
n)
J (r2i Pi)
- [
(~
f,
+
n)
{2n~(~ +An)r1ir2i}{4n~rli (1
- p.) + r 2i P1·.J
1
4] [r 1 i ( 1 - Pi) J}
- Pi) +
-
4n~r2iPi
(~+
-
(~
+ n)n r 2i P
n)E;rli (1 - Pi)}
(using
~
en),
Thus we take
(2.2.4)
2.2.3.
Three Estimators of
6.
Optimal choice of the set of weights (2.2.4) would depend on the
unknown parameter
e,
which is the ratio of the two hazard functions in
29
the PHF model.
Three estimators are proposed based on different
assumptions regarding
8.
(i)
cons tan t, 1. e., a. = a
for
1
i = l, ... , k.
This assumption, which is equivalent to ignoring the question of
weighting the estimators of
8
at each of the times of death, gives
(2.2.5)
- p.)
1
(ii)
Take
8= 1
in
{a }, i = 1 •...• k.
i
This assumption is equivalent to supposing the two hazard functions are equal at each of the times of death.
It is reasonable to sup-
pose that we will often encounter situations in which
8
is not too far
from one. or else the difference in the two survivorship functions would
be so clear as not to need experimentation.
Estimators of
e
~
and
are
(2.2.6)
and
\
A
r..r 21
·p·(r
11
l · + r 2 1·)
-1
(2.2.1)
respectively.
(iii)
Take
8
= 82
in
{a.}. i
1
=
l ..... k.
This assumption allows us to "improve" our guess of
from a specified value of
the estimators of
e
and
8.
~
b
starting
Of course. one can continue to "improve"
(see Section 2.2.4). but this iterative
approach detracts from the simplicity of the proposed estimators.
It
will be seen that iteration is often unnecessary because
8
little in value from successive values in the iteration.
With a second
iteration we have
2
differs
30
Ir2iPi(rli02 +
t:,3
Ir2i~i(rli~2' +
r2i)~1
r 2i )-1 + Irli(l -
~i)(rli02 +
r 2i )-1 '
(2.2.8)
where
8
2
2.2.4.
is given by (2.2.6).
(Conditional) Maximum Likelihood Estimator of
A fourth estimator of
6
6.
to be considered is
"
8
(2.2.9)
1 + 8"
where
is the ML estimator which maximizes the conditional likelihood
8
for the two population PHF model given by Cox (1972) (see Section 1.6,
Chapter 1).
Rewrite the conditional likelihood for all times of death under
model (1.6.3) (see also (1.7.1»,
(2.2.10)
where
Z.
l
e
i f an individual from
TIl
dies first at
t..
i f an individual from
TI
dies first at
t ..
2
Notice that the single indicator covariable
estimator
2
l
is equivalent to the
Z.
l
descr,ibed in Section 1.1, Le.,
l
i
=
Pi' Notice also that
the conditional likelihood depends on the hazard function only at the
times of death; the unconditional likelihood of a model describing times
of death
depend~
on the hazard function over the entire interval
and hence implies assumptions about
U(t)
[0,00),
over the entire interval.
In the notation of Section 1.1, the likelihood conditional on all
individuals in the two populations who are at risk of dying at
t.
l
is
31
(2.2.11)
Except for a constant the log likelihood of (2.2.11) is equivalent
to the log of the conditional likelihood given by Cox (2.2.10).
of 8,
8,
is the value of
8
The MLE
that satisfies the equation
(2.2.12)
For SSI, from (2.2.11) we see that
tic for
and
:2
is a sufficient statis-
For SSII, however, this is not true since the vectors
8.
are functions of the
,
Pi s.
'"
sufficient statistic, our estimators
and
LPi
:1
Although not functions of a
89,
and
69,
(9, =
1,2,3)
of
8
6, respectively, seem to be useful approximations to the MLE's of
these parameters.
to the MLE's.
In fact, repeated iterations give estimators tending
This is obvious, since the iterative equation given in
(2.1.12) is easily shown to be equivalent to that for the ML estimator
(2.2.12), Le.,
L
e=
P
r 2i i
rlie + r 2i
-.-->.-----'---
which gives
using
(2.2.13)
32
Also, the form of our estimators gives a rather clear picture of the
relation of the estimators to the data.
2.2.5.
~.
Nearly Unbiased (NU) Estimators of
~
Nearly unbiased (NU) estimators of
were also obtained.
Recall that
E[~]
~
-
~
+
Bias[~]
~(l - ~)2W'
using (2.2.3)
- ~[l - (1 - ~)2W]'
~
Thus
E [1 _ (l -
Each estimator
~'Iv
('Iv
1,2,3)
~)2w1 ,;,
.
~
was divided by
....
{l - (1 - ~)L.W},
with the corresponding estimators of
e
(2.2.14)
~
and
(2.2.14), to obtain the NU estimators
1,2,3).
was also divided by (2.2.14), substituting
respectively, giving
asymptotically.
"
~4
substituted into
"
8
4
and
"-
~4
for
8
~
and
since this estimator is unbiased only
The properties of these NU estimators will be compared
to the properties of the original estimators.
A
summar~
of these com-
parisons is given in Section 2.5.
Recall that Sample Scheme II (5811) denotes estimation (of
~)
in which the risk sets
R
li
and
R
2i
e
and
are no longer fixed, but are
subject to the observed natural decrement reflected by
{Pi}'
33
i
I;i
In this sampling situation
= 1,2, ... ,k.
8.
statistic for
is not a sufficient
At each time of death, with no censoring, the numbers
of individuals in the risk sets
R
1i
and
P.1 's
respectively, are related to the
R , i.e., r
2i
1i
and
r
2i
by
= r n - L P.
(2.3.1)
j<i J
r
L (1
= r 21 -
2i
~
We will now consider estimation of
2.3.1.
~1
Relationship of
Recall that
at random from
~
p.)
-
j<i
(2.3.2)
J
under SSII.
to Mann-Whitney U-statistic.
is the probability that, given one person chosen
and one from
We can show that the es-timator
TI
2
~1
dies first.
, the person from
~,under,
of
SS11, is in fact the
standard Mann-Whitney (1947) U-statistic for estimating
~.
From (2.2.5), we have
~r2iPi
~1
=
1
---~------
Lrz,p·
ill
where
k
=
r
+ r 21 , and
11
I
+
Ir
·(1 -
P.)
1 ll
i
k
represents
i
Consider first the denominator
I
i=l
D, say,of
(2.3.3)
D
?[r Z1
1
L (1 - PJ.)]Pi + L.[r 11
j <i
Using
Ii j<i
I P.J
and
1
-
L p.](l j <i J
Pi)·
34
I L pop.
i j<i J
1
we have
so that
D
=
(2.3.4)
Thus,
A
r ll r 2l + ~rl1(r1l - 1) - i(i - l)P i
(Z.3.5)
~l
r l l r Zl
The numerator of
~l
can be shown to be aU-statistic.
Let
U
the number of pairs for which
z
t(l) < t(Z), i.e., the
number of pairs of observations from
and
TIl
which the time of death of an individual in
is less than the time of death of an individual in
t(Z) .
the number of pairs for which
W
t
(1)
>
t
(2)
,
the sum of the ranks of the times of death of members
g
of
TI.
g
W
g
is the Wilcoxon rank sum statistic, g
In the notation of the estimators for
~,we
notice that
= 1,2.
35
(2.3.6)
Ul + Uz
where
= rllr Zl "
Substituting the relationship (2.3.6) in (2.3.5) and simplifying,
we obtain
(2.3.7)
U
z
Thus,
r
is the standard U-statistic for estimating
n r 21
8..
The statistic was proposed by Pitman (1948) and by Birnbaum (1956) (see
also Hollander and Wolfe, 1973, pp. 68-77).
is the minimum variance unbiased estimator of
In addition, 8.
tinuous distributions ..
1
Lehmann (1951) showed
6
6
1
over the class of con-
has all the properties of the
Wilcoxon-Mann-Whitney U-statistics, one of which is asymptotic normality
as
min(rll,r
Zl
)
+
Uz
The distribution of
r
variance
Z.3.Z.
i.e., that
00,
:
N[ r ll r Zl 8.,
6
n + r 21 +
lZr r
n 21
1
rllr2l(rll + r Zl + 1)] .
lZ
is asymptotically normal with mean
8.
and
1
Other Estimators of
6.
Although the estimators of 8. proposed in Section 2.2 were developed
for an optimal choice of the set of weights
{ail
under SSI, there was
nothing in the discussion of 8. in Section Z.l which imposed any restr±ctions on the choice of
estimator
6Z' 6
3
and
8.
1
8.'"4
{ail.
We have just seen that the unweighted
under SSII is an unbiased estimator of
6.
The estimator·s
given by (Z.Z.7), {Z.Z.8) and (2.2.9), respectively,
36
under SSII will also be considered.
four estimators of
2.4.
6
The distribution properties of all
will be presented in Section 2.5.
Extensions to Ties and Censoring.
2.4.1.
Ties.
2.1
The estimators proposed in Section
can be extended to handle
apparent "multiple deaths" (Cox's "multiplicities").
terms these represent ties.
In statistical
In practice they result from using a unit
of time to record times of death which is larger than the smallest time
between any two successive deaths.
Two estimators of
8
and
6
are proposed which are reasonable
when there are ties among the recorded times of death.
Let
r
gi
=
number of individuals from
t. , g
1
d
gi
=
=
TT
= 1,2,
at risk of dying at
1,2.
number of individuals from
g
g
i
TT
g
who die in
[t i ,t i +1 ),
= 1, ... ,k.
d.
total number of deaths observed in
1
l, ... ,k; d. > O.
1
Consider the
each of the
di
d
i
subintervals in
deaths.
[ti,t + )
i 1
Now let
number of deaths from
TT
g
in
occurred as of the time of the
Actually the superscript is
will write
formed by the times of
n.
[t ,t +1)
i
n
tb
i
which have
death, n = 1, ... ,d ..
1
but for convenience we
37
= n.
Notice that
For each of the
n
= pr{member
pin)
[ti,t i +l ),
unknown deaths in
of
TIl
death
I
r
dies first as the
li
- d
n
th
(n-l)
,r 2i
li
(n-l)
[rli - d l i
]8
Linearizing (Z.4.l) we have
n
(2.4.1)
equations of the form
(2.4.2)
We may sum the equations with respect to
n
and define
8
as the
solution of
d.1
d.
e I {[r l i
L [r 2i
n=l
n=l
(n-l) ---en)
- d 2i
]Pi '
(2.4.3)
where
---en)
Pi
1
=
{a
i f member t'f
TIl
dies first as the
n
i f member of
TI
dies first as the
n
Z
th
th
death.
death.
Simplifying (Z.4.3)·we have
(2.4.4)
(g)
where
L
is the sum over the
n = l, ... ,d.
n
death from
TI
1
subintervals in which a
occurs, g = 1,2.
(1) (n-l)
Notice that
d
is the sum over all times of death from
2i
g
L
n
TIl
of the number of previous deaths from
TI .
Z
This is in fact
V ,
2
the Mann-Whitney V-statistic defined in Section 2.3.1, which is the sum
38
of all 'pairs of death, times, one from
a member of
is
VI
TIl
dies before a member of
and one from
TI '
Z
11
Similarly.
,
2
I
j
(2)
n
d
n
which
(n-1)
li
defined in Section Z.3.l.
There are
dlid
Pr{member of
Zi
TIl
pairs of failure times; for each pair
of the pair dies first}
e
1
+
(Z.4.5)
e
If we replace the summations in (Z.4.4) by their respective expected
values, i.e.
put
I
(Z)
(n-l)
dli
n
and
I
(1)
n
(n-l)
d 2i
we obtain
O.
The solutions to (2.4.6) involve a quadratic equation in
e
(Z.4.6)
for each
i = l , ... ,k.
Combining the
weights
k
estimates of
e,
using some arbitrary set of
{a:},
we have
1
(2.4.7)
The solutions. to (2.4.7) are
e=
-1, which is obviously not possible
(and, in fact was introduced by multiplying through by
obtain (Z.4.6», and
(1 + 8)
to
39
i
o
I a d 1i(r Zi - d Zi )
i
= ----------------.--
(Z.4.8)
IaidZi(rli - d 1i )
i
An estimator for
6
is
(Z.4.9)
For the special case of
=
d.
1.
for all
1
i = l , ... ,k,both
(Z.4.8) and (Z.4.9) reduce to the respective estimators (Z.1.1Z) and
(Z.1.l5), where
d l 1..
= P.,
1.
= (1
d Z1.'
-
P.),
1.
for all
and
The choice of the appropriate set of weights when
d. > 1
1.
1.
for some
i
is left arbitrary at this time.
[ti,t + )
i l
Within each interval
we are in a SSII situation.
However, the above analysis applies whether we have a
SSII
SST or a
situation from one interval to another.
A second, more heuristic approach, for an estimator of
6
which allows for ties is now suggested.
e
and of
The form of the respective
estimators is the same as that given for the single death case.
risk sets are adjusted to some "average" number at risk
over each interval,and
A
p
= p
i
*
=
i
dli
-----~-
d
1i
+ d Zi
The
and
The estimators suggested
are
* *1. *1.
Ia.rz·p·
• 1.
1.
-
* r * . (1
La.
.1.1
l .
*
- p.)
1.
(Z.4.l0)
1.
and
,'t
* *
La.rz·p·
• 1.
1. 1.
1.
* *1. 1.*
La.rz·p·
• 1.
1.
(Z.4.11)
40
Again, the choice of the weights
*
{ex.}
is left arbitrary, and the
1
"average" risk sets are not specified.
These estimators do not reduce
to the single death estimators given by (2.1.12) and (2.1.15), respectively.
2.4.2.
Censoring.
In most follow-up studies there will be censored observations.
The exact time of censoring may be known, or the interval in which
censoring occurs may be all that is known (recorded).
Ties may also
occur among both the censored and the uncensored observations.
The expressions (2.4.8) and (2.4.9) may be modified to account for
censored observations.
We may subtract a proportion of the number of.
individuals censored in each population during each interval from the
respective numbers initially at risk in
and
at
It is also possible to find "interval" estimators of
t . , i = l , ... ,k.
1
e
and
!:,
in the sense of lower and upper bounds which correspond to extreme
assumptions about the times of censoring.
estimator we may assume the time
That is, for an upper bound
of censoring of individuals in
is at the beginning of each interval, and the time
individuals in
n
l
of censoring of
is at the end of each interval.
Reversing the
assumptions in each population would give a lower bound estimator.
2.5.
Distributions of
(£
1,2,3)
and Some of Their
and
Properties.
The discrete distributions and properties of the estimators of
given in Sections 2.2 and 2.3 were investigated when the true value of
e
is assumed known.
8'
[ .25
.50
Nine values of
.75
.90
1.0
e
were considered,
1.1
1.3
2.0
4.0], with
!:,
41
corresponding values of
=
~'
[.20000 .33333 .42857 .47368 .50000 .52381 .56522 .66667 .80000].
Recall that
e
~,
~
~
and
e
Occasionally we may refer to
= 1+8' or
interchangeably.
~
The exact distributions of the estimators of
SSI and SSII were generated for several cases.
proposed under
When the number of out-
comes in the sample space became large a sample of the possible outcomes
was drawn.
A Monte Carlo simulation is presented in Chapter IV which
investigated the large sample properties of the estimators in an approach
somewhat different than the sampling study described in this section.
2.5.1.
SSI (No .Censoring).
Suppose we observe
t
< t
l
2
< ••• < t
successive deaths at times
either from
k
1 x k
are the
k
the times of death.
for all
g~
.th
1,2, i
g
death
~
g
=
1,2, i
=
TI , respectively, at each of
2
= 1, ... ,k.
be defined such that
TI g' g = 1,2, i
Z. = g
~
There are
= 1, ... ,k.
For illustration, let
numbers at risk be given by
r
and
r'
-1
=
[r
n
r
12
r
k
13
4, and let the
r
14
]
and
r 24] , where each r gi is a specified value,
4
1, ... ,k. The 2 = 16 possible vectors are
22
2
k
Z" which represent the sequences of populations in
which deaths can occur.
r
TIl
Z' = [Zl' ... , Zk]
is from
"possible vectors
r' = [r
_2
21
and
TI , given that
2
The only restriction on the vectors is that
Let the vector
if the
or from
vectors of values representing the numbers of
individuals at risk of dying in
r . > 1
TIl
23
42
(1)
[1 1 t 1]
(2)
[1 1 1 2]
(3)
[1 1
2 11
(4)
[1 2 1 1]
( 5)
[2 1 1 1]
(6)
[1 1 2 2]
(7)
[1 2 1 2]
(8)
[2 1 1 2]
(9)
[2 2 1 1]
(10)
[1 2 2 1]
(11)
[2 1 2 1]
(12)
[2 2 1 2]
(13)
[2 1 2 2]
(14)
[1 2 2 2]
(15)
[2 2 2 1]
(16)
[2 2 2 2]
For each of the
of
2
k
possible vectors each of the four estimators
described in Section 2.2, I1 (t
11
t
lated.
= 1,2,3)
A
and
11 , can .be calcu4
The corresponding nearly unbiased (NU) estimators
11'Q, (t
= 1,2,3)
/"
and
11'
4
can also be determined (2.2.14).
ability of each of the
j
= 1, ... ,2 k
Under the PHF model the prob-,
possible sequences is calculated
from
A
I-Pi,
v.(p;8)
J
r
~
2i
(2.5.1)
]
where
Pi
{:
.th
if
1
if
1
.th
death
from
IT
death
from
IT
l
2
,
,
i = 1, ... ,k.
For any specified value of
8 we can evaluate (2.5.1) and find the
moments of the distribution of the estimators of
about zero of any estimator
The
th
Y
moment
/1, say, is
N
u
where
N
= 2k , /1 j
/1E{/1Q"I1~
'(/1;8) = E[/1Y] = I/1jv.(P;8),
y
j=l
is the estimate of
(~= 1,2,3),6
4
A
u
(2.5.2)
J ~
f or t h e J.th
sequence, and
,6 4}.
Small. Samples
When
k
is small (k ~ 8, say) it is easy to determine the exac~
distribution of each of the eight estimators proposed for
/1
under S81.
43
Five examples are given in Tables 2.1-2.5 which illustrate the small
prop~rties
sample
values of
and
O.
of each of the eight estimators of
6
for nine
The vectors of the numbers at risk in the two populations,
:2' are given in Tables 2.1-2.5.
From these examples we can reasonably conclude:
(1)
All eight estimators are biased.
(2)
There is no clear pattern as to whether each estimator overestimates or underestimates
~.
This property seems to
depend on the relative magnitudes of the numbers at risk,
and
r
2i
, at each time of death.
estimators seem to overestimate
= 1, ... ,k,
i
6
(4)
~3
1
if
:1 >:2
and otherwise to underestimate
6
for all
for at
e.
least some values of
(3)
6
All the original
usually has the smallest bias.
almost always has the smallest mean square error, MSE,
where
.
MSE[6]
= Var[~] +
2 •
Bias [6].
always has the
largest MSE.
(5)
The MSE of the original estimators is usually smaller than
the MSE of the corresponding NU estimators.
true for all values of
e,
This may not be
but in general there is little
reason to prefer an estimator which has a reduction in bias
that is frequently accompanied by a much larger increase in
the variance.
If one had more information on the influence
of the relationship of the numbers at risk it might be
possible to decide when it is worthwhile using nearly
unbiased estimators.
44
TAILE 2.1
551, EXACT 01STRlnUnON. k • b,
Exact V.tuu
e
.25
.90
.20000
MEAN
.33333
HEAN
.42857
.47368
MEAN
SO
ItSE
MEAN
SD
ItS!
1.00
.50000
MEAN
SO
HS!
1.10
.52381
MEAN
SO
·ItSE
1.30
.56522
MEAN
SO
HS!
2.00
.66667
MEAN
SO
ItS!
4.00
[4
2 I 1 II. r' - (4 4 4 4 3 2)
-2
Nearly Unbl.sed Estimators
~
A
SD
ItSE
.75
.
Original Estimators
SO
HSE
.50
!i
.80000
MEAN
SO
HS!
Al
A
2
A
3
A
4
A'
1
A'
2
h'
3
h'4
.18716
.20062
.04041
.18271
.19136
.03692
.18447
.19157
,03694
.18389
.19137
.03688
.20023
.21233
.04508
.19585
.20311
.04127
.19714
.20342
.04138
.19713
.20319
.04129
.31519
.23008
.05327
.30902
.21999
.04899
.31047
.21843
.04823
.31001
.21882
.04843
.33439
.23944
.05734
.32846
.22960
.05274
.33006
.22812
.05205
.32955
.22849
.05222
.40836
.23586
.05604
.40161
.05169
.40222
.22303
.05044
.40205
.22382
.05080
.43042
.24211
.05862
.42406
.23236
.05401
.42 /,83
.n574
.22974
.05279
.42461
.23050
.05315
.45302
.23494.
.05562
.44619
.22489
.05133
.44626
.22171
.04993
.44627
.22270
.05035
.47594
.23946
.05734
.46956
.22983
.05284
.46986
• 226M
.05145
.46971
.22710
.05186
.47924
.23338
.05490
.47241
.22340
.05067
.47212
.22007
.04921
.47227
.22108
.04964
.50250
.23684
.05610
.49616
.22729
.05168
.49605
.22407
.05022
.49614
.2150 4
.05066
.50305
.23133
.05395
.49627
.22142
.04979
.49565
.21796
.04830
.49591
.21900
• 0~874
.52652
.23381
.05468
.52026
.22435
.05035
.51983
.22099
.04885
.52003
.22201
.0493(1
.54470
.22636
.05166
.53808
.21659
.04765
.53685
.21296
.04616
.53733
.21406
.04660
.56828
.22710
.05158
.56224
.21719
.04744
.56t21
.21429
.04594
.56162
.21535
.04639
.64803
.20657
.04302
.64222
.19722
.03949
.63948
.19376
.03828
.64050
.19475
.03861
.67043
.20313
.04127
.66529
.19423
.03713
.66280
.19093
.03647
.66374
.19189
.03683
.78660
.16298
.02674
.78271
.15448
.02416
.71864
.15251
.02371
.78014
.15283
.02375
.80387
.15523
.02411
.80060
.14709
.02164
.79688
.14527
.02111
.79827
.14558
.02120
45
tABLE 2.2
551, EXACT 01STRIBUT10N, k - 7. ~i
.25
.33333
.42857
211,!:;- (4322211)
Nearlv Unbln.cd Estimntors
lI
3
A4
A'1
A'2
A'
3
lI'
4
.20723
.15576
.02431
.20679
.14408
.02080
.20826
.14353
.02067
.20774
.14356
.02067
.20097
.15295
.02339
.20035
.14131
.01997
.20178
.14078
.01982
.20128
.14081
.01983
.34324
.18988
.03615
.34258
.17587
.03102
.34366
.17469
.03062
.34328
.17500
.03072
.33516
.18865
.03559
.33428
.17464
.03050
•.33533
.17348
.03001
.33496
.17378
.03020
SO
I'.5E
.43936
.20344
.04151
.43861
.18857
.03566
.43916
.18722
.03516
.43896
.18760
.03530
.43091
.20353
.04143
.42992
.18863
. 03558
.H044
.18728
.03508
.43025
.18767
.03522
MEAN
MEAN
SO
lISE
.75
4 3
A2
SO
lISE
.50
I,
Al
A
.20000
(4
Original Estimators
Exact V.•1ues
8
-
MEAN
.90
.47368
KEAN
SO
lISE
.48461
.20707
.04300
.48383
.19200
.03697
.48410
.19065
.03645
.48400
.19104
.03660
.47620
.20777
.04317
.47519
.19265
.03712
.47543
.19130
.03660
.47534
.19170
.03675
1.00
.50000
MEAN
.51092
.20837
.04354
.51013
.19323
.03744
.51024
.19191
.03694
.51019
.19230
.03708
.50260
.20941
.04386
.50158
.19422
.03772
.50167
.19290
.03722
.• 50162
.19329
.03736
.53467
.20902
.04381
.53388
.19387
.03769
.53385
.• 19259
.03719
.53384
.19296
.03733
.52647
.21037
.04426
.52545
.19515
.03808
.52540
.19387
.03759
.52540
.19424
.03773
.57586
.20895
.04377
.57507
.19385
.03768
.57481
.19267
.03722
.57488
.19301
.03734
.56795
.21079
.04444
.56694
.19561
.03827
.56666
.19444
.03781
.56674
.19477
.03794
.67616
.20189
.04085
.67542
.18741
.03520
.67472
.18662
.03489
.67494
.18681
.03497
.66936
.20473
.04192
.66843
.19014
.03616
.66771
.18935
.03585
.66794
.18954
• 03593
.80670
.17447
.03048
.80616
.16205
.02630
.80528
.16194
.80556
.16189
.02624
.80211
.17794
.03167
.80144
.16540
•02736
.80053
.16529
.02732
.80082
.16524
.02730
SO
lISE
1.10
.52381
MEAN
SO
lISE
1.30
.56522
MEAN
SO
lISE
2.00
.66667
MEAN
SO
lISE
4.00
.80000
MEAN
SO
lISE
.02625
46
TABLE
55I, EXACT OISTR.IBUTInN. k • 7,
Exac t Values'
e
.25
.50
Orisind
I:.
.20000
.33333
'\
.50000
.52381
(;'
(;'
.20274
.13703
.01878
2
3
4
.019~5
.20440
.13678
.01873
MEAN
.34241
.19041
.03634
.15000
.18040
.03282
.35326
.IH49
.03155
.35206
.17759
.03189
.32769
.18896
.03574
.33480
.17936
.03217
.33788
.17558
.03085
.33674
.17666
.03122
.43790
.20627
.04264
.44645
.19618
.03881
.44799
.19189
.03720
.44740
.19317
.03767
.42289
.20727
.04299
.43109
.19758
.03905
.43250
.19336
.03740
.431.95
.19463
.03789
.48286
.21082
;04453
.49163
.20074
.04062
.49232
.19653
.03897
.49203
.19779
.03946
.46811
.21290
.04536
.47661
.20320
.04130
.47718
.19903
.03963
.47693
.20029
.04013
.50902
.21260
.04528
.51784
.20254
.04134
.51805
.19845
.03971
.51793
.19967
.04019
.49453
.21528
.04638
.50312
.20557
.04227
.50322
.20151
.04062
.50314
.20273
.04111
.53264
.21365
.04572
.54147
.20362
.04177
.54126
.19968
.04018
.54128
.20085
.04065
.51844
.21684
.04705
.52709
.20714
.04292
.52677
.20322
.04131
.32683
.20438
.04178
lISE
.57362
.21419
.04595
.58236
.20423
.04200
.58148
.20066
.04053
.58174
.20169
.04095
.56008
.21820
.04764
.56870
.20852
.04349
.56771
.20494
.04201
.56800
.20597
.04243
MEAN
MEAN
SO
!lEAN
SO
MlWl
HSE
.56522
(;.
.19993
.13876
50
1.30
J'l'
1
.1 ~528
.14792
.02190
MSE
1.10
J'l4
.21534
.14118
.02017
NSE
1.00
I:.~
.21708
.14091
.02015
NSE
.47368
Ne"rly Unbi.1S~'d Estimators
Estlm.ntor~
.21233
.14304
.02061
SO
.90
. (5432111]
.20114
.15252
.02331
KSE
.421157
2
(4 4 44432J. ~;
MEAN
SO
tlSE
SO
.75
1:.
!i .
2.3
!lEAN
SO
2.00
.66667
MEAN
SO
MSB
.67363
.20817
.04338
.68159
.19856
.03965
.67944
.19627
.03869
.68014
.19684
.03893
.66229
.21368
.04568
.67028
.204;:0
.04171
.. 66801
.20190
.04077
.66874
.20247
.04100
4.00
.80000
MEAN
.80442
.18095
.03276
.81018
.17246
.02985
.80753
.17232
.02975
.80840
.17213
.02970
.79703
.18712
.03502
.80290
.17855
.03189
.80014
.17843
.03184
.80104
.17822
.03176
SO
lISE
47
t'ABLE 2.4
SSI, EXACT OISTRIBlilON. k - 8.
.25
6
.20000
MEAN
SO
HSE
.50
.33333
HFAIl
SO
HSE
.75
.42857
MEAN
SO
HS!
.90
.47368
MEAN
SO
HSE
1.00
.50000
MEAN
SD
HSE
1.10
.52381
MEAN
SO
HS!
1.30
.56522
MEAN
SD
HSB
2.00
.66667
MEAN
SO
HSE
4.00
.80000
[4
3 3 3
2 1], <' - [5 5 4 3 2 2 1 1)
.2
0<1s1na1 E.tlmntors
Exact Va lues
9
!i .
MEAN
SO
HSE
Nearlv Unbln<ed Estim4tors
3
64
6'
1
6'
2
6'
3
6'4
.20096
.14184
.02012
.20305
.14112
.01992
.20231
.14114
.01992
.19533
.15479
• 023~8
.19932
.14181
.02011
.20141
.14110
.01991
.20066
.14111
.01991
.32835
.18302
.03352
.33404
.16826
.02631
.33561
.16667
.02778
.33505
.16708
.02792
.32717
.18393
.03387
,33278
.16918
.02862
.33435
.16760
.02809
.33379
.16800
.0/.823
.42266
.19218
.03697
.42889
.17705
.03135
.42970
.17521
.03070
.42941
.17574
.03089
.42202
.19347
.03748
.42824
.17834
.03180
.42906
.17649
.03115
.42876
.17703
.03134
.46747
.19382
•03761
.4737,9
.17873
.03194
.47421
.17687
.03128
.47406
.17742
.03148
.46714
.19519
.03814
.47348
.18007
.03243
.47390
.17822
.03176
.47374
.17876
.03196
.49366
.19401
.03768
.49999
.17899
.03204
.50017
.17716
.03139
.50010
.17770
.03158
.49351
.19538
.03822
.49987
.18034
.03252
.50006
.17851
.03187
.49998
.17905
.03206
.51738
.19370
.03756
.52368
.17879
.03197
.52366
.17700
.03133
.52366
.177 52
.03151
.51740
.19506
.03809
.52375
.18012
.03244
.52373
.17632
.03180
.52372
.17885
.03199
.55872
.19205
.03693
.56491
.17741
.03148
.56454
.17 574
.03088
.56466
.17622
.03105
.55901
.19335
.03742
.56527
.17866
.03192
.56491
.17698
.03132
.56502
.17747
.03149
.66042
.18189
.03312
.66599
.16838
.02835
.66493
.16715
.02794
.66528
.16746
.02804
.66131
.18280
.03344
.66696
.16920
.02863
.66591
.16797
.02822
.66626
.16828
.02832
.79518
.15304
.02345
.79915
..14212
.02020
.79774
.14170
.02008
.79821
.14168
.02008
.79639
.15308
.02345
.80044
.14207
.02018
.79904
.14166
.02007
.79950
.14164
.02006
61
62
6
.19682
.15479
.02397
48
TABLE 2.5
SSI, EXACT DISTRIBUTION, k • 7, ~i
Exact Valuea
e
.25
6
MF.AN
SO
tIS!
.50
.33333
MEAN
SO
tIS!
.75
.42857
MEAN
SO
tIS!
.90
.47368
MEAN
SO
tIS!
1.00
.50000
MEAN
SO
tIS!
1.10
.52381
MEAN
SO
tIS!
1.30
.56522
MEAN
SO
tIS!
2.00
.66667
MEAN
SO
tIS!
4.00
.80000
[5 4 3 3 2 2 Il.
:i .
[5 5 5 :. 4 3 3)
Original Estimators
6
.20000
.
MEAN
SO
tIS!
1
6
2
6
3
Nearlv Unbi.1sed F.stimatot'B
6
4
6'
1
6'
2
6'
3
t.'4
.19220
.17435
.03046
.18988
.16882
.02860
.19076
.16876
.02857
.19048
.16870
.02855
.20027
.18025
.03249
.19797
.17472
.03053
.19889
.17467
.03051
.19660
.17460
.03049
.32252
.20033
.04025
.31927
.19384
.03777
.31997
.19302
.03744
.31975
.19322
.03752
.33415
.20475
.04192
.33099
.19832
.03934
.33172
.19751
.03901
.33149
.19771
.03909
' .41669
.20598
.04257
.41310
.19908
.03987
.41336
.19779
.03935
.41329
.19816
.03950
.42987
.20865
.04354
.42642
.20187
.04075
.42671
.20059
.04024
.42663
.Z0096
.04039
.46161
.20561
.04242
.45795
.19856
.03967
.45793
.19711
.03910
.45795
.19754
.03927
.475n
.20735
.04300
.47172
.20044
.04018
.47174
.lS900
.03960
.47175
.19942
.03977
.48791
.20455
.04199
.48423
.19743
.03923
.48403
.19590
.03863
.48412
.19635
.03881
.50167
.20573
.04233
.49817
.19876
.03951
.49801
.19724
.03891
.49808
.19769
.03909
.51176
.20307
.04138
.50809
.19589
.03862
.50772
.19430
.03801
.50787
.19478
.03819
.52560
.20373
.04151
.52211
.19671
.03870
.52179
.19515
.03809
.52192
.19562
.03827
.55337
.199B
.03987
.54975
.19208
.03713
.54908
.19042
.03652
.54933
.19091
.03670
.56720
.19909
.03964
.56378
.19201
.03687
.56315
.19038
.03625
.56339
.19087
.036'.4
.65604
.18390
.03393
.65275
.17658
.03137
.6'5132
.17498
.03085
.65184
.17542
.03099
.66899
.18154
.03296
.66595
.17441
• 0304 2
.66458
.17 287
.02989
.66508
.17331
.03004
.79246
.14877
.02219
.79008
.14180
.02020
.78799
.14082
.01998
.7S876
.14096
.02000
.80229
.14430
.02083
.80015
.13752
. 01891
.79816
.13660
.01866
.79890
.13674
.01870
49
Large Samples
As the observed number of successive deaths from
TIl
and
TI
2
(k > 8, say), generation of the exact distribution of the
increases
eight estimators of
for the
6
2
k
possible sequence of populations
in which the deaths occur soon becomes economically prohibitive.
Some
results are presented which come from a preliminary sampling study of
the properties of the eight estimators of -6 using the following example.
Example:
Suppose we have observed
k
= 18 deaths from TIl and TIZ'
where the numbers at risk in the two populations at each time a death
occurred are
r'
~l
=
(2.5.3)
[10 9 8 7 6 6 6 5 544 3 3 333 2 2].
r' = [10 10 10 10 10 9 887 7 6 6 5 4 322 1].
(2.5.4)
~2
For each of the
1 •...• 2
j
18
sequences a random number
generated from the uniform distribution over the interval
E < .00085
[0.1].
If
that sequence would be selected for inclusion in the sample.
The sampling fraction
from the
was
E
.00085
was chosen so that the sample size
262.144 sequences was approximately
Eight
~stimates
of
6
200.
corresponding to each of the
populations selected for the sample were determined.
about zero were also found for
y
= 1,2.
(n)
The
n
y
vectors of
th
moments
i.e.
n .
n
\ yV A
\
A
L 6
.(P;8)/ L v.(p;8).
j
'I
J~
'lJ~
J=
J=
(2.5.5)
Five samples (of sizes 220. 224. 210, 219. and 243) were drawn for
the risk sets given in (2.5.3) and (2.5.4).
mary statistics for one of these samples.
Table 2.6 presents the sumThere is no evidence that the
NU estimators are to be preferred to the original estimators. and
6
3
50
TABLE 2.6
551, SAMPLE
01STR1DUT10~,
r' • [10 9 8 7 6 6 6
.1
~
k • 10, SAMPLE SiZE· 220
5 4 4
3 3 3 3
2J
r' - (10 10 10 10 10 9 8 8 7 7665432211
.2
Ex8C t
e
.25
Original Estlm,tors
Values
!l4
fl'
fl'
6'
6'4
.2327
.0645
.0052
.2325
.0645
.0052
.2076
.0941
.0089
.2389
.0632
.0055
.2340
.0650
.0054
.2338
.0650
.0054
.3343
.1162
.0135
.3320
.1185
.0140
.3318
.1186
.0141
.3201
.1408
.0200
.3364
.1170
.0137
.3340
.1193
.0142
.3339
.1194
.0143
.4283
.1387
.0192
.4334
.1191
.0142
.4326
.1203
.0145
.4325
.1205
.0145
.4309
.1395
.0195
.4361
.1198
.0144
.4353
.1210
.0147
.4352
.1211
.0147
.4738
.1300
.0169
.4754
.1132
.0128
.4749
.1138
.0129
.4749
.1139
.0130
.4766
.1306
.0171
.4783
.1137
.0130
.4779
.1143
.0131
.4778
.1144
.0131
SO
HSE
.4977
.1248
.0156
.4979
.1097
.0120
.4976
.1100
.0121
.4976
.1102
.0121
.5007
.1254
.0157
.5010
.1102
.0121
.5006
.H05
.0122
.5006
.1106
.0122
6
.20000
.33333
MEAN
MEAN
SO
HSE
.75
.42857
MEAN
SO
HSE
.90
.47368
MEAN
SO
HSE
1.00
.50000
~2
6
.2064
.0934
.0088
.2375
.0627
.0053
.3182
.13'99
.0198
6
5-D
HSE
.50
Nearly UnbIased Estimators
MEAN
1
3
1
2
3
1.10
.52381
MEAN
SO
HSE
.5180
.1206
.0146
.5173
.1069
.0115
.5170
.1071
.0115
.5170
.1072
.0115
.5211
.1210
.0146
.5204
.1073
.0115
.5201
.1074
.0116
.5201
.1076
.0116
1.30
.56522
MEAN
SO
HSE
.5513
.1143
.0133
.5496
.1029
.01C8
.5493
.1031
.0109
.5494
.1032
.0109
.5544
.1146
.0132
.5528
.1032
.0108
.5525
.1033
.0108
.5526
.1035
.0109
2.00
.66667
MEAN
.6276
.1023
.0120
.6253
.0937
.0105
.6256
.0951
.0107
.6258
.0952
.0107
.6308
.1022
.0117
.6285
.0937
.0102
.6289
.0950
.0105
.6290
.0951
.0105
.7213
.0776
.0122
.7180
.0708
.0117
.7220
.0752
.0117
.7223
.0753
.0117
.7243
.0773
.0117
.7211
.0705
.0112
.7250
.0749
.0112
.7253
.0750
.0112
SO
HSE
4.00
.80000
MEAN
SO
MSE
51
usually has the smallest MSE.
It seems reasonable that the estimators
are biased for large samples as they were shown to be for small samples.
For these five examples, however, we are unable to distinguish true
bias from bias introduced by sampling variability.
It was of interest to test two null hypotheses
H6 l ):
(2) .
Ho .
A beta distribution describes the distribution of the
estimator
11,
A Johnson
SB
(2.5.6)
distribution describes the distribution
of the estimator
(2.5.7)
11,
Two test statistics were calculated.
Using the method of moments,
the parameters of the beta (Elderton and Johnson, 1969) and the
distributions (Johnson and Kitchen, 1971) were estimated.
SB
These esti-
mates are available in manuscript.
A Kolmogorov-Smirnov (KS) one-sample test statistic was calculated
for each estimator at each of the nine values of
were used which are appropriate for large samples.
e.
Critical values
The test is con-
servative when the parameters are unknown and must be estimated from the
sample data.
There is evidence to reject both
Chi-squa~e
and
(2)
Ho
.
goodness of fit tests using ten intervals of width
were also calculated.
.1
These test statistics, each with seven degrees
of freedom, also rejected both
and
(2)
Ho
.
It is not necessarily surprising to find these two continuous
distributions fail to closely' represent the discrete distributions of
the estimators of
11.
Although the sample sizes on which the test
statistics were calculated are reasonably large (about 200 observations
52
per sample), the number of distinct values the estimators can assume
is much smaller. Under SSI the number of deaths which occur in
k
is a sufficient statistic. There are only (k + 1) distinct
i=l ~
I p,
64
values which the ML estimator
{6£,6i
other estimators
n)
distinct values.
(£
Until
1,2,3)}
k
64) may
(and hence
assume.
also assume less than
The
2k
(or
becomes very large no simple continuous
distribution is likely to closely approximate the distributions of the
estimators.
2.5.2.
SSII (No Censoring).
Suppose that from some initial time
individuals from
k
=
r ll
+
r 2l
and
TI
we follow
r
ll
and
r
2l
, respectively, until all of the
individuals die.
ordered times at which the
period.
2
to
k
Let
t
l
< t
2
< '"
< t
k
represent the
deaths are observed during the follow-up
The numbers at risk of dying in each population at the time a
death is observed are random variables, i.e.,
r'
_1
r
r'
_2
r
lk
2k
],
],
where
r
ll -
Ip·,
j<i J
p,) .
J
The vectors
r gl , g
=
1,2,
and
depend on the number initially at risk,
and on the populations from which deaths have occurred
previously. As in Section 2.5.1, let ,the vector
each element
Z.
~
=
g
if the
i
th
death is from
Z
be defined such that
TI, g
g
=
1,2,
53
l, ... ,k.
NC
There are
possible orders in which the
deaths from the two populations could occur.
= 2.
[:j
The
= 6
As an illustration, let
possible orders of the populations from
which the four deaths can occur, and the corresponding risk sets, are
(1)
[1 1 2 2]
r'
_1
[2 1 0 0]
[2 2 2 1],
(2)
[1 2 1 2]
r'
_1
[2 1 1 0]
[2 2 1 1],
(3)
[122l}
r'
_1
[2 1 1 1]
[2210],
(4)
[2112]
r'
_1
[2 2 1 0]
[2111],
(5)
[2 1 2 1]
r'
_1 = [2 2 1 1]
(6)
[2 2 1 1]
r'
_1
r'
_2
[2 2 2 1]
[2 1 1 0],
[2 1 0 0],
Notice that the sequences of risk sets used for examples in SSI are in
fact possible sequences for SSI1.
However, in general this is not true
for all 551 sequences.
Estimates of
and
r
that
~l
.for all possible NC sequences for any value of
can be found using the four estimators described in
2l
Section 2.3,
~
~~ (~
=
"
1,2,3) and ~4.
~.
is an unbiased eStimator of
sider nearly unbiased estimators of
A
~4
It has been shown (Section 2.3.1)
because one of our estimators,
~
~l'
It was decided not to con-
corresponding to
~2'
~3
and
was unbiased.
Small Samples
The small sample properties of the four estimators under 5511 were
investigated by generation of their exact distributions which depend on
the possible orders of death in the two populations, for
k
= r ll +
r
2l
~
10.
Four examples are given in Tables 2.7 and 2.8 to
illustrate their properties.
From these examples we can reasonably
54
TABLE 2.7
Ie· 7, r
SSII, EXACT DISTRIBUTIONS:
Exact V"luas
8
.25
E5timator~ •
6
.20000
'l..
MEAN
SO
MSE
.50
.33333
MEAN
SO
HSE
.75
.42857
MEAN
SD
MSE
.90
.47368
MEAN
SD
MSE
1.00
.50000
MEAN
SD
MSE
1.10
.52381
MEAN
SD
MSE
1.30
.56522
MEAN
SO
MS!
2.00
.66667
MEAN
SD
MSE
4.00
.80000
MEAN
SD
MS!
ll
• 3, r
• 4 nnd Ie • 8, 'II • 4, r
21
21 • 4
Ie • 7, ell • ), e 21 • 4
6
2
6
3
6.
4
Estimators, k • 8, ell • 4
lI
1
11
2
lI
3
e
21
6
.4
4
.20000
.18856
.03556
.19267
.17812
.03178
.20235
.17694
.03131
.20091
.17701
.03133
.20000
.16832
.02833
.18997
.15573
.02435
.19907
.15496
.02402
.19763
.15507
.02405
.33333
.22361
.05000
.33033
.22138
.04902
.33737
.21452
.04603
.33633
.21548
.04644
.33333
.20242
.04097
.32486
.19704
.03890
.33159
.19132
.03661
.33058
.19221
.03695
.42857
.23434
.05492
.43106
.23522
.05534
.43430
.22632
.05125
.43382
.22758
.65182
.42857
.21396
.04578
.42445
.21'248
.04517
.42758
.20465
.04188
.42712
.20582
.04236
.47368
.23585
.05562
.47896
.23697
.05618
.48019
.22778
.05193
.47999
.22908
.05252
.47368
.21616
.04673
.47213
.21552
.04645
.47330
.20726
.04296
.47313
.20848
.04346
.50000
.23570
.05556
.50686
.05602
.50690
.22744
.05177
.50687
.22873
.05236
.50000
.21651
.04688
.50000
.21600
.04665
.50000
.20767
.04313
.50000
.20890
.04364
.52381
.23493
.05519
.53205
.23536
.05546
.53102
.22637
.05130
.53114
.22764
.05187
.52381
.21622
.04675
.52522
.21561
.04649
.52416
.20733
.04299
.52431
.20856
.04350
.56522
.23213
.05389
.57565
.23128
.05360
.57282
.22285
.04972
.57320
.22404
.05026
.56522
.21439
.04596
.56899
.21307
.04541
.56613
.20515
.04209
.56655
.20633
.04257
.66667
.21731
.04722
.68077
.21122
.04481
.67421
.20547
.04227
.67512
.20628
.04262
.66667
.20242
.04097
.67514
.19704
.03890
.66841
.19132
.03661
.66942
.19221
.03695
.80000
.17847
.03185
.81345
.16460
.02727
.80476
.16430
.02702
.80600
.16432
.02704
.80000
.16832
.02833
.81003
.15573
.02435
.80093
.15496
.02402
.80237
.15507
.02405
.2365~
55
TABLE 2.8
ssn,
!.'Ulc:t
e
.25
EXACT
6
!lEAN
SO
HSE
.50
.33333
MEAN
SO
HSI
.75
.42857
MEAN
SO
KSI
.90
.47368
MEAN
SO
KSI
1.00
•.50000
MEAN
SO
HSI
1.10
.52381
MEAN
SO
HSI
1.30
.56522
MEAN
SO
KSE
2.00
.66667
MEAN
SO
KS!
4.00
.80000
k· 9, r
Estimators. k • 9, r
V~lue"
.20000
DISTRIB~llONS:
MEAN
SD
HSE
U
• 4,
• 5 end k • 10, r 11 • 5, r
• 5
r 21
21
• 4, r
ll
61
62
h
.20000
.16193
.-02622
.19233
.14968
.02246
.33333
.19293
.03722
21
• 5
Estimators, k • 10, r
ll
. 5.
r
21
.5
3
64
.19110
.13580
.01852
.19956
.13496
.01821
.19807
.13518
.01828
.33333
.17888
.03200
.32551
.17160
.02958
.33175
.16713
.02793
.33079
.16795
.02821
.43140
.19420
.03772
.42857
.18921
.03580
.42472
.18552
.03443
.42762
.17910
.03208
.42120
.18009
.03243
.47168
.19478
.03795
.47749
.19591
.03840
.47368
.19118
.03655
.47223
.18823
.03543
.47331
.18146
.03293
.47316
.18H7
.03330
.50430
.20225
.04092
.50437
.19469
.03792
.50435
.19582
.03836
.50000
.19148
.03667
.50000
.18866
.03559
.50000
.18183
.03306
.50000
.18284
.03343
.52381
.20353
.04143
.52944
.20138
.04059
.52849
.19394
.03763
.52862
.19505
.03807
.52381
.19123
.03657
.52513
.18831
.03546
.52414
.18152
.03295
.52429
.18254
.03332
.56522
.20123
.04049
.57298
.19813
.03932
.57032
.19113
.03656
.57070
.19219
.03697
.56522
.18959
.03594
.56876
.18604
.03462
.56609
.17955
.03224
.56648
.18054
.03260
.66667
.18856
.03556
.67809
.18127
.03299
.67184
.17646
.03116
.67278
.11726
.03146
.66667
.11888
.03200
.67449
.17180
.02958
.66825
.16713
.02793
.66921
.16795
.02821
.80000
.15492
.02400
.81106
.14171
.02022
.• 8.0272
.14123
.01995
.80410
.14139
.02001
.80000
.14847
.02204
.80890
.13580
.01852
.80044
.13496
.01821
.80193
.13518
.01828
h
4
hI
h2
.20158
.14860
.02208
.20003
.14878
.02213
.20000
.14847
.02204
.32863
.18746
.03516
.33537
.18188
.03309
.33432
.18279
.03341
.42857
.20267
.04108
.42875
.20040
.04016
.43187
.19309
.03730
.47368
.20416
.04168
.47647
.20235
.04095
.50000
.20412
.04167
3
h
56
conclude, in addition to
(1)
1':,.
(Z)
In the special case
is an unbiased estimator of
1
1':"
that
r
= r
ll
21
e
All estimators are unbiased for
(i)
. (ii)
underestimates
I':,Z
e
for
6
I':,
for
e
< 1
=
1.
and overestimates
The same appears to be true for
> 1.
63
"-
and
(iii)
The MSE for estimators is at a maximum when
As
(3)
1':,4' though for these estimators the bias is slight.
r ll + r Zl
k
estimates
I':,
it appears that
(4)
Whether
1':,.
/:I.Z
1':,3
and
"-
1':,4
overestimates or under-
seems to depend on the value of
relationship of
1.
=
increases the MSE decreases.
For the case
always overestimate
e
e
and the
and
In all cases of
smallest MSE (although often only slightly less than that
of
When
and
are very different from each
other this was not always found to be true.
The distributions of the estimators under SS11 are also discrete.
However, the NC estimates corresponding to each possible point in the
sample space are usually distinct.
statistics were calculated to test
for
l':,e{l':,l
(1
= 1,Z'~)'~4).
Chi-square goodness of fit test
(2.5.6) and
As before, the test statistics were
calculated using ten equal intervals, each of width
the beta and
SB
(Z.5.7)
.1.
Parameters for
curves were determined from the first two moments of
the exact distributions.
From Table Z.9 there is evidence to suggest
that in this case the beta distribution and the
SB
distribution are
57
TABLE 2.9
CHI-SQUARE COODNES5 OF FIT TEST STATISTICS. EXACT DISTRIBUTION • SSII
Exact
Va1uea
!'
t.
.25
.20000
.33333
.75
.42857
• ~O
.90
.47368
1.0
.50000
1.1
.52381
1.3
.56522
2.0
.66667
4.0
.80000
k • 7 <3.4)
t.1
t. 2
A
3
A
4
SB
B.. ta
133.60
28.91
8.08
4.70
4.20
4.67
7.75
34~08
197.76
204.26
29.53
6.80
3.46
2.96t
3.40
6.40
35.59
373.00
SI
'eta
214.11
33.94
7.21
2. n t
1.69tt
1.58tt
3.41
24.35
193.27
6.57
2.20t
1.17tt
1.09tt
2.92 t
25.83
391.76
2.51 t
2.24 t
2.54 t
2.28t
2.01 tt
1.75tt
2.04 tt
2.41 t
5.23
4.93
30.95
33.86
225.85
477.52
2.43 t
5.20
30.56
224.57
1. 79 tt
2.16 t
4.91
33 •.35
473.69
403.63
5.
'eta
153.03
5.
leta
36.30
26.43
27.57
5.71
5.42
152.86
26.26
234.69
27.32
5.72
5:43
235.89
**
2.14 t
u
**
**
k • 8 (4.4)
Al
A2
A
3
A4
S.
leta
376.33
64.21
13.06
5.43
4.29
5.23
11.34
64.21
376.35
693.31
68.56
11.86
4.33
3.22 t
4.13
10.34
68.56
693.40
2.31 t
3.37
10.66
2.38 t
9.59
79.29
&8.37
681.02
1840.80
62.47
461.16
944.21
681.02
79.29
12.44
3.60
'eta
1839.68
88.37
11.39
2.61
t
1. 36 tt
5.
leta
461.16
10;68
3.43
2.81 t
2.35
1. 17 t+
3.23'
943.72
62.47
66.64
SB
'eta
495.85
65.18
10.55
t
1.84 tt
2.76t
1044.61
71.14
10.15
1.48tt
2.37 t
**
**
5.
9.94
t
2.96
t
2.58
...
2.62 t
9.22
8.50
66.64
9.02
8.61
65.18
495.85
71.14
1044.88
**
••
Ir. • 9 (4,5)
Al
A
2
SI
leta
51
leta
748.17
126.43
••
22.50
1510.22
136.00
1503.45
169.49
. 19.55*
.*
26.36
4689.09
181.98
23.41
••
6.96
4.66
6.70
20.54
1094.23
3.98
17.35
•
•
148.08
4.31
2.07 tt
161. 17
2812.~0
7.02
3.55
4.85
18.01
•
160.21
1727.00
4.76
1.59tt
2.98 t
16.07
181.61
6474.90
21.22 •
18.34 •
177 .85
1860.34
199.34
7278.40
148.11
22.98 **
6.24
3.86
6.09
160.68
20.50
4.17
1.82tt
3.87
A
3
SI
'eta
57.21
3242.01
A4
58
leta
57.57
146.20
22.53 **
5.83
3183.93
159.00
20.32
3.95
**
**
•
•
3.38
1. 52 tt
5.48
20.17
•
172.29
1806.46
3.44
17.51·
193.09
7012.39
••
**
Ir. • 10 (5,5)
B
leta
2332.08
322.32
50.87
322.32
44.48
3.59
12.96
7.78
43.42
354.47
13.95
8.72
8.53
6335.88
37.16
354.48
113.03
6338.67
62
5.
Beta
4549.10
20345.34
399.02
457.34
52.79
11. 94
8.27
6.09
2.67 t
10.87
7.24
44.42
48.21
39.90
399.02
45/;35
4549.10
20521. 33
A
3
S.
Beta
5135.67
450.14
60.95
16.11
9.72
14.94
51. 73
450.14
5135.67
4.22
9.08
43.99
508.97
24630.48
Al
5
24408.24
508.93
52.93
10.18
4926.88
435.05
59.20
9.23
14.33
50.23
435.05
4926.88
491. 82
..
15.48
23200.52
51. 72
9.90
4.06
8.83
43.00
491.86
23327.32
A
A4
5.
Beta
..
**
tt ( 2
2
Pr X9 < X9 , .01 • 2.088) • . 01
tpr{x~ < X~ •• 05 • 3.325} • . 05
**
*
{2
2
..
••pr X9 < X9 •• 95 • 16.919} • . 95
pdx'9 < X'9, .99 • 21.667} •• 99
When the ay.bol appl1ea to all value. In a column, it is given at the bottom of tho'1t column.
**
58
reasonable forms to describe the distributions of the four estimators
6
of
e
when
is close to
1.
In each case the beta distribution
gives a better fit than the
SB
distribution (as judged by the
values).
Large Samples
k, the total number of deaths from
As
and
TIl
increases,
generation of the exact distributions of the estimators soon becomes
economic~lly
prohibitive.
study, using as an example
Some results from a preliminary sampling
r
11
erties of the four estimators of
Example:
r
6
Zl
= la, of the large sample propwill be reported.
individuals from
Suppose
respectively, are followed until all 20 die.
NC
~ [~~]
number
E
~
E
and
For each of the
possible orders in which che deaths can occur, a random
was generated from the unit uniform distribution.
If
.0011 that order would be selected for inclusion in the sample,
and the risk sets
evaluated.
sample size
r, g
~g
1,2,
The sampling fraction
n
was approximately
corresponding to that order were
.0011
200.
was chosen so that the
Due to the symmetry of the
distribution of the possible orders of death when
r
l1
= r
Zl
only half
of the set of combinations was generated; the other half was obtained
from the complements of the orders of death of the first half.
For each of the
of
6
n
orders selected for a sample, four estimates
corresponding to that order were determined.
about zero for Y= 1,2 were found
sizes
using (2.5.5).
186, 170, 192, 184 and 222) were drawn.
summary statistics for one of these samples.
confirm those described for small samples (6
The
th
Y
moments
Five samples (of
Table 2.10 presents the
The conclusions generally
1
is naturally no longer
59
TABLE 2.10
SSIl, SAMPLE DISTRIBUTION, k • 20, r 11 • 10, r 21 • 10
SAMPLE SIZE· 186
Exact Values
e
.25
E~tim"tors
A
.20000
MEAN
SO
HSE
.50
.33333
MEAN
SO
MSE
.75
.42857
MEAN
SO
MSE
.90
.47368
MEAN
SD
HSE
1.00
.50000
MEAN
SO
MSIl
1.10
.• 52381
MEAN
SO
HSE
1.30
.56522
MEAN
SD
MSE
2.00
.66667
MEAN
SD
MSE
4.00
.80000
MEAN
SD
HSE
Al
A2
A3
A4
.1921
.0809
.0066
.1838
.0735
.0057
.1879
.0733
.0055
.1875
.0730
.0055
.3354
.1335
.01711
.3282
.1275
.0163
.3315
.1259
.0159
.3311
.1261
.0159
.4386
.1245
.0156
.4353
.1194
.0143
.4368
.1170
.0138
.4366
.1173
.0138
.4781
.1209
.0146
.4769
.1157
.0134
.4774
.1132
.0128
.4774
.1135
.0129
.5000
.1203
.0145
.5000
.1151
.0132
.5000
.1125
.0127
.5000
.1128
.0127
.5198
.1208
.0146
.5209
.1156
.0134
.5204
.1130
.0128
.5205
.1134
.5557
.1239
.0154
.5587
.1187
.0141
.5574
.1163
.0136
.5575
.1166
.0137
.6646
.1335
.0178
.6718
.1275
.0163
.6685
.1259
.0159
.6689
.1261
.0159
.8079
.0809
.0066
.8162
.0735
.0057
.8121
.0733
.0055
.8125
.0730
.0055
.01~9
60
unbiased in this restricted set, although it is actually unbiased).
The null hypotheses
tested for
~E[~t
(t
H(l)
o
(2.5.6) and
"
= 1,2,3)'~4]'
Kolmogorov-Smirnov test statistics
the nine values of
e.
(2.5.7)
were
Table 2.11 gives the one-sample
D
n
for each estimator at each of
The parameters were estimated from the sample
using the method of moments.
These are available in manuscript form.
The KS test statistics are conservative when the parameters are unknown
and must be estimated from the sample.
They are also conservative when
the true distribution is discrete (see Gibbons, 1971, pp. 85-87).
is evidence to reject both
for
e = 1,
or
~ =~.
for all values of
and
e,
There
_except
It remains uncertain whether a test statistic
more appropriate for the beta and the
SB
distributions with parameters
unknown would lead to the same conclusions for
e=
1.
Modifications of
the significance limits of the KS statistic along the lines of the
research of Stephens (1974) or of Durbin (1975) should be possible when
the underlying distribution~ are either a beta or a
As yet, however, such results are not available.
SB
distribution.
61
TABLE 2.11
lOLHOGOROV-SMIRNOV ONf, S~WLE TEST STATISTICS /,0 0n
5511, SAMPLE OISTRI~UTION , SA!'lPLE 5IZE • 186
.~6522
2.0
.66667
4.0
.80000
1. 3672 t
2.9907
7.0131
12.0306
.4985
1. 3515
2.9H8
6.9389
12.0673
.6419
1. 5091 t
3.2601
7.6430
12.3255
.6123
1.4720
3.2045
7.6000
12.3904
.6961
1.5602
3.3087
7.6924
12.3666
3.2551
7.6508
12.4303
Exact
Valuea
9
6
.25
.20000
.50
.33333
.75
.42857
.90
.47368
.\
Sa
12.0306
7.01H
3.2319
'1.4601 t
.5269
BeU
12.0673
6.9389
3.1811
1.4389 t
Sa
12.3255
7.6430
3.5397
1.6073
6
2
6)
6
4
1.0
.50000
t
1.1
.52381
1.3
t
aeta
12.3904
7.6000
3.4813
t
1.5692
Sa
12.3666
7.6924
3.5880
1.6581 tt
.6676
1. 5246t
.6962
1.5604t
3.3089
7.6944
12.3604
.6676
1. 5246 t
3.2551
7.6526
12.4239
aeta
12.4303
7.6508
3.5318
1.6216t
Sa
12.3604
7.6944
3.5883
1.6583tt
Beta
12.4239
tt
7.6526
tt
3.5317
tt
1.6216
t
t
tt
tt
ttpdlii On > z .01 • 1. 63} • .01
tpr{1ii On> z.05 • 1.36} • •05
When the symbol applies to all values in a column, it is liven at the bottom of that column.
tt
CHAPTER III
SOME ANALYTICAL AND El-IPIRICAL INVESTIGATIONS OF
THE HAZARD RATIO FUNCTION (HRF)
In Chapter II we considered the special case when the ratio of two
hazard functions is constant in the form
8,
or more conveniently in the form
We called this model the proportional hazard functjon (PHF) model.
In this chapter we wish to consider a more general situation
when the ratio of two hazard functions is a function of
t.
Thus We
introduce the hazard ratio function (HRF)
8(t)
for all
t,
(3.0.1)
and the hazard delta function
6{t)
for all
t.
(3.0.2)
Sometimes we may write (3.0.1) as
]Jl(t;h)
8(t) = - - - - - - ]J2(t;S2)
where
~'
-g
[t:-g 1 ... "'gm
c 1
is the
(3.0.3)
1 x m vector of parameters for the
63
hazard function of
n, g
g
= 1,2.
We will discuss the behavior of the HRF for some situations
when
(i)
population mortality data,
(ii)
empirical data, and
(iii)
parametric models with known forms of the hazard functions
~ (t;~
g
), g
-g
= 1,2,
are used.
The special case of the HRF defined by the proportional hazard
function model has frequently been assumed by both actuaries and demographers to describe the relationship between mortality experience in
two populations.
In particular, the PHF model is assumed to hold in
many statistical tests for comparing survival curves (for example see
Peto and Peto (1972) and Crowley (1974».
The brief overview assessment of the HRF in this chapter is
motivated by a problem
effect of
and
3.1.
'~'
8(t)
to be considered in Chapter IV, that is, the
not being constant for all
t, on the estimation of '8'
under the assumption that the PHF model is valid.
The HRF for Some Population Mortality Data.
Four examples of the HRFfor some population mortality data are
presented in Figures 3.1 to 3.4.
The life table data are taken from
Keyfitz and Flieger (1971), pp. 354-372.
Let
to < t l <
intervals of width
...
< t
w-l
denote a sequence of non-overlapping
h. = ti+l - t.
1
1
and midpoint
The usual approximation to the hazard function at
t
~
=
1
t
~
1
1
= - -h 9.np.,
.
1
1
1/(
t.1
2.
+ t i +l )·
is
(3.1.1)
64
where
qi
[t i ,t i +1 )
is the conditional probability of dying in the interval
given alive at
t
i
, provided
qi
is small (less than
.3,
say).
The values of
e(t~)
1
i
o, ... ,w
- 1
(3.1.2)
were plotted in Figures 3.1 to 3.4 for all age intervals up to age 85.
The four examples are
(i)
Males and females, United States 1966 population,
(Figure 3.1).
(ii)
Nonwhite and white males, United States 1966 population
(Figure 3.2).
(iii)
United States (males) and England and Wales (males), 1967
population (Figure 3.3).
(iv)
United States (males) and West German (males), 1967
population (Figure 3.4).
From these illustrations we can see that the HRF's are not simple
functions.
In particular, they cannot be regarded as effectively con-
stant, except over very restricted ranges of
3.2.
t.
.'!'lJe HRF for _~ome Empirical Data.
Three examples are presented of the HRF estimated from some
empirical data reported in previously cited references.
A brief descrip-
tion of these examples is given.
Example!:
The data are one year results ofa clinical trial reported
by Freireich et ~~. (1963) which compared a treatment against a placebo
in the maintenance of remissions in acute leukemia patients.
There are
65
FIGURE 3.1
MALES AND FEMALES, UNITED STATES 1966 POPULATION
(Keyfitz
a~d
F1ieger (1971), p. 354)
•..-r---'-----------------------,
·
i
i
;'
............
...
.." ·
......
i
::
i
·i
l-
..
iI.1
...
. ..
•••
.... ....
AGE III YEARS
FIGURE 3.2
NONWHITE AND WHITE MALES, UNITED STATES 1966 POPULATION
(Keyfitz and F1ieger (1971) , pp. 356, 358)
·
•.
::
i
!'
:
... ,.r:
"
...
.........
... ·
...
..."
I
C
I
·i
l-
.... .... ....
•••
.... ....
AG! III YURS
11.'
M.'
...
III.
66
FIGURE 3.3
UNITED STATED (MALES) AND ENGLAND AND WALES
(~ffiLES),
1967 POPULATION
(Keyfitzand F1ieger (1971), pp. 360, 472)
::,...---------------------------,
::
::
..
.....
::
..."
."
....
....
---.....
"
'"
~
::
..
..
i
,.
t.
....
.... ....
t ••
1t,1
iI.'
...
II.'
.II.'
I",
FIGURE 3.4
UNITED STATES (MALES) AND WEST GERMAN (MALES), 1967 POPULATION
(Keyfitz and F1ieger (1971), pp. 360, 438)
.it ' . . . - - - - - - - - - - - - - - - - - - - - - - - - - .
.
::
.
:
b
...
.......
....---
":I
..
..." ::
...'"
M
:
~ to
It.'
....
It.'
.... ....
AGE IN YEARS
....
ll.•
".f
....
,It.
67
21 individuals in each population.
Gehan (1965a) "and Cox (1972).
The data have been reanalyzed by
Cox concluded (graphically) that an
exponential distribution was a reasonable model for the lifetime data
in each population.
Thii naturally implies that the PHF model is rea-
sonable for comparing survival experience.
Table 3.1 presents a life
table analysis of the data and Figure 3.5 is a plot of the estimate of
8(t).
TABLE 3.1
EXAMPLE 1, SAMPLE ESTIMATES OF
HRF
(Data taken from Freireich et a1. (1963»
Interval
[ti,t i +l )
7T
l
:Treatment Std. Error
1
~
- hi inPli (Approx.
Est. )
7T
2
:Placebo Std.Error
1
- ~inP2i (Approx.
Est. )
1
weeks
Est.
HRF
Std.Error
8 (t ~ )
(Approx.
Est.)
1
[0,4)
.00000
.00000
.06798
.00000
.0000
.0000
[4,8)
.05427
.02719
.07192
.03608
.7545
.1566
[8,12)
.01786
.01787
.17329
.07217
.1031
.0409
[12,16)
.02175
.02176
.17329
.10206
.1255
.0552
[16,20)
.02634
.02635
.10137
.10206
.2599
.1212
[20,24)
.08412
.05976
.00000
.00000
68
FIGURE 3.5
EXAMPLE 1, PLOT OF ESTIMATED
(Data taken from Freireich
~!
HRF
al. (1963»
WEEKS
Example 2:
The data are some preliminary results from a cooperative
"clinical trial which compared six maintenance regimens for remission of
·acute leukemia.· The data were made available by Breslow (1974a) with
permission of the Children's Cancer Study Group A.
He noted that an
exponential distribution is a reasonable approximation to the estimated
remission curve.
Table 3.2 presents a life table analysis of two treat-
ment populations formed by pooling regimens 1, 2 and 4 and regimens 3
and 5, with 152 and 116 individuals, respectively.
estimated HRF is given in Figure 3.6.
A plot of the
69
TABLE 3.2
EXAMPLE 2, SAMPLE ESTIMATES OF
HRF
(Data from Breslow (1974a), the Children's Cancer Study Group A)
Interval
[ti,t i +l )
TTl:Regimens
Std.Error
1,2,4
1
- hiJl.nPli
(Approx.
Est.)
TTZ:Regimens
Std. Error Est. HRF Std. Error
3,5
1
- --Jl.np
h.
2i
1
(Approx.
Est. )
e(t~)
(Approx.
Est. )
days
[0,lZ0)
.00069
.00020
.00125
;00031
0.5502
.0019
[120,240)
.00186
.00035
.00301
.00055
0.6188
.0017
[Z40,360)
.00176
.00039
.00208
.00054
0.8482
.0027
[360,480)
.00229
- .00051
.00155
.00055
1.4759
.0055
[480,600)
.Q0214
.00062
.00292-
.00104
0.7320
.0034
[600,720)
.00125
.00063
.00205
.00119
0.6101
.0045
[720,840)
.00056
.00056
.00140
.00141
0.3963
.0052
[840,960)
.00250
.00177
.00000
.00000
70
FIGURE 3.6
EXAMPLE 2, PLOT OF
ESTI~~TED
HRF
(Data from Breslow (1974a), the Children's Cancer Study Group A)
:
._-----------------------------,
:
It.'
'It. I'
",0
".
n.'
10,0
10.
'10
".0
DAYS
Example 3:
The
dat~
are some preliminary results from a clinical trial
reported by Glasser (1967) of survival following surgery for lung cancer.
The two populations are. those with "low" ratios of vital capacity
to predicted vital capacity and those with "high" ratios.
and 95 individuals respectively in the two populations.
There are 36
Glasser also
states that an exponential distribution provides a reasonable fit to the
survival curve.
Two life table analyses of the data are given in Tables
3.3a and 3.3b with the respective plots of the estimates of
in Figures 3.7a and 3.7b.
e(t)
given
The two analyses are presented to indicate
the effect of grouping on the estimate of the HRF.
71
TABLE 3.3a
EXAMPLE 3, SAMPLE ESTIMATES OF
HRF's
(Data from Glasser (1967»
TIl: "Low"
ratios Std.Error
TI : "High"
2
ratios Std. Error Est.HRF Std.Error
1
- h~nPli
i
(Approx.
Est.)
1
- ~~nP2i
[0,30)
.00838
.00297
.00332
.00111
2.5250
.0416
[30,60)
.00251
.00178
.00241
.00098
1.0414
.0294
[60,90)
.00600
.00301
.00088
.00062
6.8414
.1906
[90,120)
.00000
.00000
.00206
.00103
0.0000
.0000
[120,150)
.00202
.00202
.00292
.00131
0.6927
.0265
[150,180),
.00468
.00331
.00196
.00113
2.3926
.0738
[180,210)
.00000
.00000
.00320
.00160
0.0000
.0000
[210,240)
.00445
.00445
.00361
.00180
1.2340
.0484
[240,270)
.00545
.00545
.00099
.00099
5.5173
.2559
[270,300)
.01352
.01361
.00222
.00157
6.0797
.2456
[300,330)
.00000
.00000
.00238
.00168
0.0000
.0000
[330,360)
.00283
.00200
[360,390)
.00287
.00287
Interval
[t i ,t i +1 )
1
(Approx.
Est.)
6(ti)
(Approx.
Est. )
days
72
TABLE 3.3b
EXAMPLE 3, SAMPLE ESTIMATES OF
HRF's
(Data from Glasser (1967»
Interval
[t i ,t i +1 )
TIl: "Low"
ratios Std.Error
1
~
- hi 9, nP 1i (Approx.
Est.)
TI : "High"
2
ratios Std.Error Est. HRF Std.Error
1
~
- ~9,nP2i
1
(Approx.
Est.)
8(t~)
1
(Approx.
Est.)
days
[0,50)
.00504
.00179
.00319
.00085
1.5793
.0149
[50,100)
.00530
.00217
.00162
.00066
3.2736
.0378
[100,150)
.00121
.00121
.00205
.00084
0.5916
.0135
[150,200)
.00317
.00224
.00253
.00104
1. 2495
.0217
[200,250)
.00575
.00408
.00267
.00120
2.1544
.0378
[250,300)
.00579
.00581
.00192
. 00111
3.0128
.0716
[300,350)
.00000
.00000
.00145
.00103
0.0000
.0000
.00444
.00257
[350,400)
73
FIGURE 3.7
EXAMPLE 3, PLOTS OF ESTIMATED
HRF's
(Data from Glasser (1967))
(a)
--- - - - - - - - - - - - - - ~
<>
o
o.
8J.O
l~O
.• ' O.
3~O.
\ 00.
DAYS
(b)
"'-r------C>
C>
C>
C>
o.
RO.O
Ii,Q.
DAYS
'00.
74
These three examples each include censored observations (in which
the censoring is not necessarily the same in both populations being compared).
=
An actuarial estimator of p ., g
g~
1.2, which uses the concept
of "effective" number initially exposed to the risk of dying in an
interval was employed (Benjamin and Haycocks, 1970).
~
hazard functions
g
=
=
1,2; i
1
~
--£np .
(t ~ )
g
h.
~
g~
~
O, ... ,w- 1
and the HRF
were then calculated.
Estimates of the
£nPli
8(t~)
~
An approximation to the
variance of the estimated hazard functions is given by
Var[~ (t~)]
g
where
E.
[ti,t + ).
i l
the risk of dying in
individuals in
TI
~
g
TI
g
(3.2.1)
2~
h.p .E .
~ g~
gl
is the "effective" number of individuals in
g~
uals in
__ . qgi
TI
g
exposed to
It was taken to be the number of
who survived the interval, plus the number of individ-
who died during the interval and the proportion of the
interval in which the individuals censored were observed.
tion to the variance of the estimated
An approxima-
8(t!), using the first order
~
approximation to the variance of a ratio of random variables, is given by
Var[e(t~)] ~
(3.2.2)
~
Since the hazard function is a rate (Elandt-Johnson, 1975),
usually described in terms of events per person time units, we must
group the data in order to estimate it.
As can be seen pictorially
from Figures 3.7a and 3.7b, the interval lengths' which determine the
groups are an important factor in the resulting estimate of the HRF.
Also, the sample size and observed number of deaths affect the precision
of the estimate of the HRF.
This suggests that we will have a difficult
task in using empirical results to make conclusions about the form of
75
the HRF.
If, however, the data fit parametric distributions, then the
functional forms of the hazard rates are known, and the behavior of their
ratios can be investigated analytically.
3.3.
The HRF for Some Parametric Models.
In this section we derive analytic forms of the HRF for some
parametric distributions.
The examples selected are models frequently
used in the description of biomedical lifetime distributions.
Table 3.4
presents a summary of eight of these models (Barlow and Proschan, 1965;
Johnson and Kotz, 1970; Ord, 1972; Patel, 1973; Mann, Schafer and
Singpurwa11a, 1974).
For convenience we wilf use IHR (DHR) to describe
hazard functions which are monotonic increasing (decreasing) functions
of time.
Those hazard functions which are IHR (DHR) for t < T an DHR
(IHR) for
t > T
will be called unimodal.
For illustrative purposes we
~(t)
will deal with parameter values of the models for which
all
t
E
< 1
for
[0,100].
It is useful to take note of the following properties of hazard
functions.
Remark 1:
Effect of transformations on hazard functions.
positive continuous random variable with density function
hazard function
~T(t).
fo~mation of
such that
T
Let
Y = geT)
T
g-l(y)
~T(g
-1
exists.
d
(y) )dy[g
and
be a
and
T
-1
It is easy to show that
and
Y
is given by
(y)].
In particular, consider the linear transformation of
location and scale parameters of,
fT(t)
T
be a monotonic increasing trans-
the relationship of the hazard functions of
=
Let
0
0.3.1)
T with
respectively, T =oy + f,.
76
1-AnI.E 3,4
SOttE rArJ,Hl:l'laC .·AHTLJES OF
Family
HaZJlrd l'unction
t > 0
lJ(t) > 0
[11~'fI:rl\l'Tl"\S
~U::::I~-:I'~~-l~(:;I'-t::,£t-::::=l=- Comm~nt"
P(l)-l-F(t)
1
1
lJ(t;A) - A,
1. F.xponent1al
). > 0
(~)(~)
lJ(t;a,te) -
-
K-I
U;
"" .\
K
,
.\
I
-----+------------+------------t
2, We1bull
(Ell-treme
Value,
Type Ill)
1J)
('XP[-At]
exp[-(n) )
a,te > 0
1111'"
- ---------1-------
UI _
Ct-' J'(K-J+l)
u, -
Ct-'IF(2K-'+1)
WhC'lT
DHR
J'('f)
...:=-] exponential
~"3, 6 nnd 0°1 is
oJmi l.tr in shape
to .1 norm.,l
distribution.
1I1R(~
th~
iL
DHR(K
gilmm,1 fillictic!n
> 1)
3: 1)
------+--------------t-----------+-------------+-------3. Hakeh8lll
+ Be tet ,
1'(t;A.,8,o;) - A
10.,8 > 0
Compertz
(A-O is
Compertz or
Extreme
Value
Type 1)
~xp(_(B/K)(e~t_l) - At)
Go:np,'rtz (A ~ 0)
iu'arrroximHtion
to 11dbull for
large x.
1l1R (K > 0)
DIIR(K
0)
Sec AI'I" ",11K A1
3:
4. Unear
lJ(tla,l3) - a
Hazard
Distribution
+ Bt
CI
> 0
exp(-(Clt + I,at'))
JIIR(ll > 0)
Dlln(B --;; 0 such
lh.1t
< II for
Sec- ArI"'lIdix A2
at
(see ltodlin
all t). -
(1967) )
-....:--:..:--+--------------1-----------f--------------- - - - - - - - 5. LOlno....al
lJ(t;~.<1)·
(~<1 J• {11- t
-
u,
r
H----<-)}
£nt _
(¥)
~ c;:llt
u, -
J ,
?r
~.
>
+ 1,0')
0"
('
1..,2
(,.
~~t
JIIR. tlo lOUR
(tinit"odal)
-1)
a > 0
- - - - - + - - - - - - - - - - - - + - - - - - - - - - ----- --~----_r__----6. Truncated
1J(t;t,B) •
Logistic
~
-g-»)) -I •
~xp(Ull)
t,B > 0
7. Gamm,l
(Pearson
Type 111)
11 ~).p [
If(t;a,B) t a- l
t
e- t/a
a,B > 0
1--=---=----- ,
a
f(a) e [l-F(t) J
t
where 'F(t) •
Jx
(I-I
1 _
Jo
a-I
-x/0
_x_ _'_'__ dx
rca)
a"
u,
U2
lJ(t;o,B)
_~-1
1
+
t
Cl
",
~e1 r:xponu\~
"f
ll-m/2, ~~2 1r
chi-square
00'
IHR(et > 1)
DlIH (ll
1)
3:
-
,
[1
a,a
+
t"J- B
til
~
U2
~
]+(l»)
OR.[r.-(l)
u '
0
f'[',[e-(~) •
'1
> 0
beL.,
--l'---
DIIR(rl < 1)
~ot JIIU(fl > 1).
l+(I)J
i:~ unllnod.:ll w1lh
mod£' n l t fHlCh
thnt t _ (a-I)-(I
0
- u:
\oll}(~J·'· !~[P.'l]
_ _ _ _ _ _-'
IlIR
dx
foa
8. Burr
[see Burr
(1942) J
!,ppu,d Ix A3
-!i/' II
-xIs
~
I'(fl)
0
-------i
~~x)(r~
{,XPI- .I:.;1'J'1
£:.:
• - - - - - - I s('~
{ 13[1 + exp(- t-f;
.
is
lht>
rClll:!lon [SI'£.'
Ilur.- a"d GJ,J;,k (196B)]
.
-'-
_
77
The hazard functions are related by
~(y)
Remark 2:
< t <
a~(ay
+
~).
(3.3.2)
Effect of truncation on hazard functions.
FT(tla ~ T ~ b)
-00
=
00,
Let
be the cdf of a continuous random variable
over the restricted range
FT(tla -< t ~ b)
T,
a < T < b, i.e.
0
for
a <
FT(t) - FT(a) ,
for
a < t < b.
for
t > b.
to
FT(b) - FT(a)
1
The hazard function of the truncated random variable is only affected by
truncation on the right, i.e.
llT(t\a < T < b)
fT(tla ~ T ~ b)
(3.3.3)
=---~--,------:-::-
1 - FT(t\a < T < b)
f ( t)
T
(3.3.4)
3.3.1.
The HRF for Some Parametric Models from the Same Family of
Distributions.
A brief review of the HRF for pairs of hazard functions from some
parametric models will be considered.
sented in Table 3.5.
A summary of this review is pre-
For illustrations we will restrict ourself to
selecting pairs of hazard functions which are monotonic increasing whenever possible.
Notice that the delta hazard function,
increasing (decreasing) when the HRF
(decreasing).
e(t)
~(t),
is monotonic
is monotonic increasing
78
TABLE 3.5
SUHHARY OF Tm: IlEIIAV!'JR 01' 'IIIE IIRF FOR SOl-IF. PAIIJ\If1:T1UC HODELS ntOll TilE SAtfE FANILY OF 1JISTRIBUTIONS
==----"""'1~~=--~=
-·-----,--....,-'10:"-'t-,-,n-!"'c--,---,-I\·-,-,·-,-,h'.~-~ ~"_!!!~X_0:)_(...t'.L)----r--.-,.-077<"---,-;-;--;;-T.:1c---
L
Exponential
2. Wei bull
C~_o.:..r,'"",,.t:c:;''',o.:..t=__ -----!..!lsy(>a~ I nJt
t
For all values
of t
A1nA,=O
3. tbkl'ham
Gompe"rtz
K 1 ""K 2
K,
:Jnd
or
Tr !vi',l
Do::'r re;lS in·
No
KI = K,
Ut,-l1'-".:.."'c:.d::a.~l:'-
No
KI <
> K,
K2
AI"A,=o
and
and
"",
(Xl
A
No
A,nA,=O
K, >
C<l~e :
11m O( t)
11m o(t)
as t -~ _- ' _
"9 t -. 0
t--,-;.:c-"_-"_-If-"''--'--_
t·tOl\O! otllr
I
.:.F.:.:o""'' '!' 'lJ-.v
No
K1
>
K,
'"
K' <
K,
0,
.
B, =B,=O
A; +
-[Adl,K,
+(",-Kd
K1
> K,
KI < K,
.
K1 > K,
0 I( 1 < K,
AlIA, •
K, < K, < 0
'"
Al + B,
(A,R,KI
K1 < K2
.
0,
0'
8,
1
B,B,}
> 0
4. Linear
No
or
BI
~
a.-
0,
TrlvLll C.1se
5. Logno",al
l:}'~,-i l' iL"a Z.
.
evidence
(i)
~,
~,
and
B::P'l-I·t~tl{..
f!vidf!nce
(1) ~I = ~, and
",
~,
and
Pos';ihle for
<
'1,
or
snIT','
(11) ~I <
~,
and
of par ..... T'leter
0, > 0, or
(11) ~I >
Lbfmcat..
p-J}i"eyr~e
0, < 0,
01 > 112
flln,. ticn
O.
01 < 0,
m.
°1 > 0,
0, 0,
v""ltles.
and
m
.
~ I
-
0,
>
~,
£i
oj
0, - 0,
and
~ 1
------+-------+-------+--------I-----~
rrivL1! case
6. Truncated
LogIstic
~. =
r"
III
8,
Md
III
1;1 > 1;,
1;1 < 1;,
H·,piri<!ai
ev~Jenco
7. Gllmm4
0,
~
0,
~
I
. 8,
GI = 8, and
Possth]l'
Sl1l1l~
of
for
function
I)'-ll''-l,"".t~r
v.1.1u("s.
t' 'OJ' ""
81
",a
E'n,:-':.rl,-'al.
1(' for
Possihlc for
rt)~5i1)
some function
~;
No
0111.-'
s~,ts
(
No
p,\r.'ln'1eter
,.. here
" I
'I)
81
,~
v
8. Burr
[l+exp(~:) J
ec'idcHce
of p.,\rallleter
vnlueA.
~,
R, [l+exp(,g) 1
L\nd
evidence
<
1-._-----+------
1~'I.-'
and
:. ' or
\'~~ r~,1.
Alw,I'yS
O.
'" ,
""
('II
B,
01
> 0,
< ""
[~~r .
.11 • "2
1,
I~'
fll > 0,
(11 <
('I,
a
0.3,
(1,87
.
the v,llue
of O(t)
at t
1.
.
79
3.3.1.1.
HRF for Two Weibu11 Distributions.
From Table 3.4 we have
l-ll(t)
e(t)
= l-l2(t) =
=
[:~w~~]
C't
where
K =
[:~][:~:]
and
l-l(t;C't ,K )
1 1
l-l(t;C't ,K )
2 2
d
K -K
t 1
2
C't,K > 0,
g g
g
1
= 1,2,
,
=
Kt
d
= K1
(3.3.5)
- K2 ·
1
The HRF is monotonic increasing (decreasing) for
(K
<
l
K
When
2 )·
d
1
> K
2
t·, this
is zero the HRF is a constant for all
includes the special case
distributions.
K
K
1
= K 2 = 1,
the HRF for two exponential
The three general shapes of the HRF for two Weibull
distributions are shown in Figure 3.8 (i.e., where d < 0, 0 < d < 1,
and
d > 1).
3.3.1.2.
HRF for
Tw~
Makeham Gompertz Distributions.
From Table 3.4 we have
A + B e
1
1
8(t)
A + B e
2
2
KIt
Kzt
A B > 0,
g' g
g
= 1,2.
The HRF is an increasing (decreasing) function for
and
Al
of
Al
=
A2
e (t)
=
A
Z
= O.
0
and
(3.3.6)
K >
l
K
The HRF is constant for all values of
K
1
=
K
Z'
with respect· to
{AZB l Kl - [A1 B2 K Z +
(K
2
-
(K
2
t
l
<
K
2
)
when
From consideration of the first derivative
t, the HRF is unimodal i f
K >
2
~)B1 B2 J} > 0, (see Appendix Bl).
K
1
and
A special
80
FIGURE 3.8
HRF
FOR PAIRS OF WEIBULL DISTRIBUTIONS
~l
(t;50,1.5)
and
~2(t;60,2.0)
* (t;60,3.5)
and
u* (t;50,2.2)
2
~l
g
g
U)(t)/U2(t)
(d < 0)
.
::
g
!:
Q"
I
i
ig
'oj
•
•
udt)!u,(t)
::
U2(t)!II, (t)
I:
(0 < d < 1)
50 I
.,"'
1>0.0
IO.
10.0
'0.0
,00
case of the Makeham distribution is the Gompertz distribution
(A = 0),
i.e., an extreme value type I distribution truncated from below at
t = O.
A simple relationship exists between a Weibull and a Gompertz distribution.
If
T
has a Weibull distribution
Y
=
~nT
has a Gompertz
distribution.
The HRF's for two sets of parameter values are given in Figure
3.9.
'81
FIGURE 3.9
HRF
~1
FOR TWO MAKEHAM GOMPERTZ DISTRIBUTIONS
(t;.03,.OOl,.03)
and
~2(t;.02,.OOl,.OS)
.~-----------------------,
it
.::
I:"
iP
;I
h
I
:L----~<:::::::.
-+----,..-,,-.•--.'•.-.--...-.•-....,.,.-.•- -,-••-.•--....-..-.•--,-,•.-.--'.. .- --'-,,-.,--\,".
T1~
3.3.1.3.
HRF for Two Linear Hazard Distributions.
From Table 3.4 we have
(3.3.7)
8(t)
The HRF is monotonic increasing (decreasing) for
>
~~]
S
2
It is a constant for
~
=0
and for
The HRF's for two sets of parameter values are given in Figure
3.10.
.
82
FIGURE 3.10
HRF
FOR TWO LINEAR HAZARD DISTRIBUTIONS
~1 (tj.005,.001)
and
~2(tj.004,.002)
::
.~
:
:
~-----------'--'--lJl~(t:lJ--2-(t-)
,. ..
----1
---------------10,0
10,'
10.'
'11,0
50.0
61,'
n~
3.3.1.4.
70.0
1'.0
to..
IIG.
HRF for Two Lognormal Distributions.
From Table 3.4 we have
B(t)
(3.3.8)
°2 M(z2 )
0.3.9)
°lM(zl) ,
where
<Hz )
g
pdf with
function
z
g
is the standard normal cdf, ¢(z ) is
the corresponding
g
£nt - t;g
, and M(z ) is the reciprocal of the hazard
ag
g
common~y
called the Mills Ratio, g
1,2.
83
The behavior of the HRF for two lognormal distributions is difficult to assess analytically.
From (3.3.9) we see that the HRF for two
lognormal distributions (each of which is unimodal) is of the same form
as the HRF for two normal distributions (each of which is monotonic
increasing).
methods.
The behavior of the HRF was investigated by some graphical
The parameter values for the pairs of hazard functions we con-
sidered are given in fable 3.6; Figure 3.11 illustrates the HRF for one
of the pairs of parameter values.
It appears that the HRF is monotonic increasing (decreasing) for
and
01 > 0z
and
(01 < 0Z)
01 > 0Z).
~1 >
and also for
sz
and
°1 < 0z
For some function of the parameter values the
HRF may be unimodal or have more than one point of inflection.
3.3.1.5.
HRF for Two (Left) Truncated Logistic Distributions.
Recall from Remark Z in Section 3.1 that truncation on the left
(i.e., t
~
a
in this case) does not affect the hazard function.
From
Table 3.4 we have
8(t)
t
where
yg
-
~
8g
g
,
~g,8g >
8 Z[1 + exp(-yz)]
81 [1 + exp(-Yl)]
a
for
g
=
,
(3.3.10)
l,Z.
~l
The HRF is constant only in the trivial case of
8
1
=
8 ,
2
respect to
From consideration of the first derivative of
t
(or more conveniently, of
log 8(t»
the HRF is monotonic increasing (decreasing) for
(~l < ~Z)
(see Appendix BZ),
It is unimodal when
=
~2
8(t)
and
with
we can conclude that
81
= 8 Z and
~l > ~2
81 f 82 ,
The HRF's for two sets of parameter values are given in Figure
3.12.
84
TABLE 3.6
PARAMETER VALUES OF THE HRF FOR PAIRS OF LOGNORMAL DISTRIBUTIONS
1T
1T
l
Comment on
2
e (t)
1':1
°1
1':2
°2
(1)
0.0
1. 30
0.0
1. 35
monotonic
(2)
0.0
1. 30
0.0
1.40
monotonic
(3)
0.0
1. 50
0.0
1. 55
monotonic
(4)
0.0
1.20
0.0
1. 50
monotonic
(5)
0.2
1.00
0.2
1. 50
monotonic
(6)
0.5
1. 30
0.5
1. 35
monotonic
(7)
0.5
1.60
0.0
1. 75
monotonic
(8)
0.25
1. 75
0.0
1. 80
monotonic
(9 )
0.0
1.00
1.0
1.00
>
1 inflection
(10)
0.3
1.00
0.2
1.00
>
1 inflection
(11)
0.5
1.00
1.0
1. 00
>
1 inflection
(12)
1.5
1.00
1.0
1.00
>
1 inflection
(13)
0.0
1.00
-0.5
1.00
>
1 inflection
(14)
0.0
1. 20
0.5
1.20
>
1 inflection
(15)
0.0
1.20
1.0
1. 20
>
1 inflection
(16)
0.0
1. 30
1.0
1. 30
>
1 inflection
(17)
0.5
1. 30
1.0
1. 30
> 1 inflection
(18)
0.0
1.50
0.5
1.50
> 1
(19)
0.0
1. 30
1.0
1.40
>
(20)
0.0
1.00
0.5
1.50
unimodal
inflection
1 inflection
85
FIGURE 3.11
HRF
FOR TWO LOGNORMAL DISTRIBUTIONS
,
i
,.
.
!l
=,.
,8
.
~ ~
1,- -----_._------ -----------
~
~-f"--~'I.-'
..-~:a.•--,...'.-a--,-.a-.--,..r:-.;--,..-c~-.;--~~~-.--,..
.. -.--,-:.-,-:....
FIGURE 3.12
HRF
FOR TWO TRUNCATE]) LOGISTIC DISTRIBUTIONS
~1 (t;10,5)
and
~2(t;15,10)
,.
g
~dt)l1J2(t)
lO.1
'"
.
to .•
""
...
• Itt:
" ..
TO.'
"
.
"'
lOt.
86
3.3.1.6.
HRF
for Two Gamma Distributions.
set)
a g ,8
. g > 0,
0.3.11)
where
g
=
rea g )
1,2.
t
is the gamma function and
F (t)
g
J
x
(a g -1) -x /S
e
dx,
nn
a
°
rca g )8 g g
A special case is the exponential distribution
the chi-square distribution (a
=
2v,8
=
2) with
v
(a
1), and
degrees of freedom.
The HRF is difficult to assess analytically and is investigated
using graphical techniques.
parameter values
An example of the HRF for a set of
is given in Figure 3.13.
For all cases considered it
appears that the HRF can be either monotonic or unimodal.
parameter values such that
and
a
l
< a
2
6
and
it is unimodal.
1
>
8
2
For some
or
In other cases it is
monotonic.
3.3.1.7.
HRF
for Two Burr Distributions.
From Table 3.4 we have
-a
8(t)
=
a16l[1
derivative of
B(t)
is always unimodal,
t
2]
,
-a
a262[1
The HRF is constant for
+
+
1]
t
a g ,6 g > 0,
g
=
0.3.12)
1,2.
From consideration of the first
with respect to
t
it can be shown that the HRF
(see Appendix B3).
The HRF's for two sets of parameter values are given in Figure
3.14.
87
FIGURE 3.13
HRF FOR TWO GAMMA DISTRIBUTIONS
;I-·~-·-----·-····--··_--·-----·-
I
::
~l (t)/IJ1 (t)
--------
..
ro.'
I'.'
".0
.'0
1.0'
".0
10,0
CO
~
...
,
TABLE 3.7
PARAMETER VALUES OF THE HRF FOR PAIRS OF GAMMA DISTRIBUTIONS
TI
TIl
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(Xl
61
2
5
3
6
4
8
3
5
5
10
4
10
3
5
1
6
2
5
5
2
3
3
5
6
2
2
3
5·
5
3
6
5
4
3
3
3
6
5
·6
3
3
(9)
5
(0)
4
8
4
(11)
(12)
(13)
(14)
(15)
(16)
Comment on
2
6
6
5
4
cx
2
3
6
2
6
e (t)
2
1
3
3
6
2
4
1
2
3
4
4
4
unimodal
unimodal
unimodal
unimodal
monotonic
monotonic
monotonic
monotonic
monotonic
monotonic
monotonic
monotonic
monotonic
monotonic
monotonic
monotonic
88
FIGURE 3.14
HRF
~1
FOR TWO BURR DISTRIBUTIONS
(t;4.0,.2)
and
~2(t;1.5,.5)
:
w
;;
~I
II
:
::
"-:
V
0
it
•.
3.3.2.
'"
".
...
n,o
..
'
to.
'fO,O
''''
The HRF for Pairs of Parametric Models from Different Families
of Distributions.
One usually finds it necessary to assume that the survivorship
functions of the two populations being compared are represented by the
same parametric model with different parameters.
Of course, it is
possible to have situations in which the hazard functions for
7f
2
are from two different parametric families of distributions.
and
In
particular, when the shapes of the hazard functions are very different
we may find it unreasonable to attempt to describe the lifetime distributions by the same parametric model.
As we have seen in Section 3.3,
the form of the HRF, even for pairs of hazard functions from the same
family of distributions, is not necessarily simple to describe.
The
HRF for pairs of hazard functions from different families of distributions would often be even more difficult to describe.
sider any of these possible cases here.
We will not con-
CHAPTER IV
ESTIMATION OF
UNDER SOME NON-PHF MODELS
~
In Chapter II we considered the special case, called the PHF
model, where the hazard ratio function (HRF)
assumed to be constant for all
t, i.e. 8(t)
8(t)
=
~l (t)/~2(t)
= 8 for all
t.
is
We
A
derived some estimators
~
=
8/(1 + 8)
~l' ~2' ~3
and
~4
of the parameter
in this. model (Sections 2.2 and 2.3).
For simplicity of
A
notation, throughout this chapter we will use
~4
as equivalent to
so that when we drop the subscript there is no confusion.
mators
~
are
fun~tions
of the numbers at risk in
populations from which a death occurs.
n
l
and
~4'
The estin
2
and the
The importance of the PHF model
lies in its frequent use for comparing and for estimating survival
functions.
In Chapter III we investigated the more general situation in which
the HRF is a function of
t.
When we can specify the parametric family
of the failure distributions, the HRF may be a well-behaved function of
t, and in some cases a constant (see Table 3.5).
For many life table
models we found that the HRF is not constant, except perhaps over a
restricted range of values of
t.
The use of empirical evidence to draw
conclusions about the form of the HRF may be difficult, especially for
small samples.
Thus, the results presented in Chapter III suggest that
the PHF assumption may be valid only under certain conditions on the
parameter values or on the range of values of
t.
We need to ascertain
90
what is the "effect" of a non-PHF model (1. e., a nonconstant HRF) on the
~
estimation of
under a PHF assumption.
In most of the chapter we will assume that we can
parametric family of the failure distributions.
know the HRF
8 (t)
spl~cify
the
We will therefore
and the hazard delta function
In Section 4.2 we will present the results of a Monte Carlo
simulation study which was designed to compare the effects of a non6~,
constant HRF on the estimators
~
= 1, ... ,4.
In Section 4.3 we
will suggest some alternatives to the PHF model and outline methods for
assessing the adequacy of the PHF model with respect to these alternatives.
4.1-
The Estimator
Function
6
as an Estimator of an "Average"
--
6 (t) .
For any experiment let
(ordered) times of death.
we then know
of- the
- Delta
6(t)
t
l
<
...
<
t
k
denote
Assuming that we know
k
WI (t)
at each of the times of death.
observed
ana
w2 (t)
If we assume
that the PHF model is valid, we obtain an estimator
6
of a "6".
This is a constant that may be regarded as some "average" of the
functional form of this average.
d(t;p)
We will consider the function
(4.1.1)
A[lI(t)] - 1I
as a measure" of the deviation of this average from the estimator
It is apparent that the distribution of
d(t;p)
6.
in repeated sam-
pIes would be extremely difficult to determine. It depends on the joint
91
distribution of the random variables
Mt.) , i
t:., which is a function of the set
estimator
theory we can derive the distribution of
1, ... ,k, and the
=
l.
under
{Pi}
Mt) .
In
d(t;p), but in practice it is
almost impossible.
However, we suggest an approximation to the expected value of
the estimators, t:,t t
= 1, ... ,4,
with regard to the underlying failure
distributions, and consider the deviation measured by the difference
between the approximation and
t:.t'
To be more specific, we take
Ia.r2·p,
• l.
l. l.
= E
(4.1.2)
l.
[.La.rz·p· + La.r ·(1
i l. l. l.
i l. 1 l.
La.r
·E[p.]
. l. 2 l.
l.
_ _ _ _ _---::l.:..-
where
_
(4.1.3)
k
I
represents
L
{ail
is the set of weights corresponding to
i=l
i
the estimator
and
t:"
r
. Mt.)
.i l .
l.
(4.1.4)
D
i
i
with
8(t.)
l.
=
t:,(t.)/[l - 6(t.)]
l.
1
and
Explicitly, the four forms of
set of weights
are given below.
{ail
D.l.
=
[r 1.
2
A1[~(t)]
l, ... ,k,
+ Mt.)(r
·
l'
l l.
- r 1')]' .
Z
which correspond to the
for each of the four estimators
t:,t' t
= 1, ... ,4
92
(4.1.5)
(4.1.6)
(4.1.7)
where
(4.1.8)
The corresponding deviations are
~=1,
... ,4.
(4.1.9)
We also consider some rather simple weighted averages of the form
L w(t.)Mt.)
11
A[Mt)] =
where
w(t.)
1
i
I
i
(4.1.10)
w(t.)
1
is a weight which is possibly a function of
t.
93
for all
The simplest type of weight would be
i
= 1, ... ,k.
This gives
(4.1.11)
with corresponding measure of deviation
(4.1.1Z)
A second type of weight which does depend on
function of the
numb~rs
times of death, such as
at risk of dying in
w(t )
i
=
TIl
t
is one that is a
and
TI
rlirZi/(rli + r Zi )'
f
I
Z
at each of the
This gives
rlirZiL\(ti)
r l i + r Zi
rlir Zi
r li
(4.1.13)
+ r Zi
with corresponding measure of deviation
(4.1.14)
Weights such as those suggested for
the extreme values of
~(t),
weights which reflect
the
A3[~(t)]
are reasonable if
occurring at early values of
t, are given
larger numbers of individuals at
r~sk
of
dying during the initial follow-up period.
A Monte Carlo simulation study is described in Section 4.Z, which
was performed to investigate
(i)
comparisons among the estimators
6~,
~
=
1, ..• ,4, for some
nonconstant HRF models, and
(ii)
comparisons among the three "averages" of the
L\(t.)'s
1.
94
which are assumed to be close to the estimators
9, = 1, ... ,4.
4.2.
Mont~
6£,
'
Carlo .?imulation Study.
Recall the following notation
r
-g
the
k x 1
vector of values representing the number of
individuals from
the
k
TI
who are at risk of dying at each of
g
times of death.
the hazard function of
J..l g (t;t,;gl,t,;g2)
parameters
t. = time of the
1
t,;gl
.th
1
and
t,;g2' g
TI
g'
indexed by
= 1,2.
f, ... , k.
death, i
We considered only two parametric models for the hazard functions
of
TI,
g
the Weibull and the Gompertz distributions.
Using any suitable
random number generator for exponential distributions we can generate
(random) times of death for these two distributions by an appropriate
transformation.
In particular, if
X
is a random variable with a unit
exponential distribution, then
T
=
.1/ t,;2
t,;lX
(4.2.1)
has a Weibull distribution and
(4.2.2)
has a Gompertz distribution of the forms given in Table 3.4 where
~l
= aI'
~2
=K
and
we considered the
distributions.
~RF
~1
=B
and
~2
=K
respectively.
For this study
for pairs of distributions from the same family of
We used the random number generator program VARGEN, which
95
is in the library of the Computer Science facilities at the University
of North Carolina;
4.2.1.
SSI Simulation Study.
For Sample Scheme I we let
the number at risk in
r'
_1
=
and
and
TI
2
:2' the vectors of values of
, respectively, for
=
k
[20 19 19 18 18 17 17 16 15 15 15 14 13 12 11 10
7
7
7
7
7
7
6
6
6
6
6
5
5
4
4
3
deaths be
40
9
9
8
8
3
2
1
1],
(4.2.3)
and
[20 20 19 19 18 18 17 17 17 16 15 15 15 15 15 15 15 14 14 13
r'
_2
13 12 11 109
8
8
7
6
6
5
4
4
3
3
2
2
1
1
1].
(4.2.4)
At each of the deaths
T ..
g1J
=
t ..
g1J
=
= 1, ... ,k)
we generate a time of death
~g(t;~gl'~g2)
from the distribution with hazard function
for each of the
j
(i
1, ... ,r ..
g1
r.
gl.
individuals at risk of dying, g
The time of the i
th
1,2, i
= 1, ... ,k,
death is given by
(4.2.5)
We require that
(4.2.6)
Tg1J
.. > t.1- l'
i.e. each time of death for individuals at risk of dying at the i
th
death is greater than the observed time of the (i - l)th death, with
We then determine the observed values of the
to the definition used throughout, i.e.
P. 's
1
according
96
p~
Given the vectors
the estimates of
t.
is from
if
t.1
is from
= {Ol
1
:1' :2' and the resulting vector
~£'
~
if
£
=
1, ... ,4
- which were derived in
Section 2.2 under the assumption of a constant HRF.
~(t.),
1
the delta function at each of the
sample, i
and
=
1, ... ,k.
A3[~(t)]
t.
1
We then find
times of death in this
Al [~(t)], A [~(t)]
2
,Calculation of the "averages"
are straightforward from (4.1.5)-(4.1.8), (4.1.11) and
(4.1.13), respectively.
and
'" we obtain
p,
d3(~'~)
The values of the deviations
dl(~'€)' d2(~'~)
follow immediately.
These steps were repeated for 200 samples.
'4.2.2.
Discussion of SSI Simulation Study.
A summary of the first "two sample moments of the four estimators,
for
k = 40
deaths and
and.
given by (4.2.3) and (4.2.4),
respectively, and for three choices of the parametric forms of
~g(t;~gl'~g2)
is given in Table 4.1.
Plots of the respective delta
functions are given in Figures 4.1-4.3.
Also included in the figures
are histograms representing the frequency distributions of the 200
sample estimates given by the four estimators, and the sample means.
Note that the histograms have a vertical scale as a base.
From these results we may reasonably conclude the following.
(i)
The estimators
~2' ~3
and
~4
their first two sample moments.
and
(ii)
~l
~4
are similar in regard to
The distributions of
~3
are also very similar.
always has the largest variance.
When the true delta
function is monotonic decreasing (increasing)
~l
tends to
97
TABLE 4.1
SUMMARY OF 200 SAMPLE ESTIMATES OF "6" UNDER SOME
SPECIFIED PARAMETRIC MODELS, SSI
Model specified by two Weibu11 distributions,
(a)
~1(t;50,1.5)
Mean
and
~2(t;60,2.0),
SD
k
= 40
{A2[~(t)]}
{A3[~(t)]}
d2(~'~)
d/~,~)
(.5478)
(.5925 )
d1(~'~)
.5956
.0848
..5924
-.0032
-.0478
-.0030
.5384
.0778
.5810
.0426
.0094
.0542
.5387
.0787
.5489
.0102
.0091
.0539
.5387
.0787
.5323
-.0064
.0091
.0538
Model specified by two Gompertz distributions,
(b)
~1(t;.005,.05)
and
~1(t;.008,.04),
k
= 40
(.4779)
(.4535)
.4576
.0933
.4535
-.0042
.0203
-.0041
.4694
.0910
.4502
-.0192
.0085
-.0159
.4689
.0906
.4777
+.0087
.0090
-.0154
.4689
.0907
.4736
+.0047
.0090
-.0154
(c)
PHF model specified by two Weibu11 distributions,
~1(t;50,1.5)
and
~2(t;50,1.5),
k = 40
(.5000)
(.5000)
.5022
.0914
.5000
-.0022
-.0022
-.0022
.4998
.0892
.5000
.0002
.0002
.0002
.4994
.0894
.5000
.0006
.0006
.0006
.4994
.0894
.5000
.0006
.0006
.0006
98
H
tr.l
tr.l
..
tr.l
Z
0
H
E-l
~
H
~
H
:f
H
/
I
I
(I)
~
~
;
:;
H
~
I
~
/:
/ :
,-"
/:
I,
~
~
~1
H
N
~
.
'-0
tr.l
+oJ
'-"
H.
H
rz..
,f-
I_,/'/
/
.
L_.~,_." .~~ -'! ._- :._-,. . S
L,
i
I
I
L._
•
j
lUJ
N
;:1
,....,
+oJ
~
t.?
0
.
.
P-I
tr.l
-:t
r-..
0
U
~
J
,.r
H
~
I
'"d
p
<]
m
.?'
,.<
r-..
~
.-l
.<]
~,
0
.
lI')
0
.
.-I'
.,
In
+oJ
'-"
1-
.-l
;:1
tr.l
~~
g
H
rJ)
~
ril
.-1
~
U'l
0
0
N
~T.<
0
tr.l
"'"'
~
P=<
t.?
0
E-l
tr.l
H
IJ:l
, -,
FIGUllli 4.2
HISTOGRAHS OF 200 SA.:.'rPLE ESTIHATES OF
~l
(a)
K"I:':EX
t"
(t;.OOS •• OS)
t" (t) IS SPECIFIED 3Y T\-;O GO~1PERTZ DIS'I:RIEUTIONS, SSI
and
~2(t;.~a8,.a4)
~:
('J)
"2
I"
I
II
L
I
~
''''f
~ -------------;;;;.:-----1~,
i"
i
r--'~
I
LI
Y
I
i
~.,I
I
,
I
I
~
~o
4"
tI~r.
(e)
0'
.,
;,.
~,
i.
I
~
~ ~lmm
-~
I
~ • 'I
,
.....;
!,.:~.-! :~~
t'
I
,
L.
..
L
i
I
I
!-
:0
,:,
;
,
(r.)
rI
II
T"'
I
~
~J
r
L-_.-_--
I
,I"
~
~
f'~
~L, ~~;.;::;:,-----L
i
I
Lj
I
!
I
,.'
,:i
~----c,;;;~<~
r---.r'1
r"'
.,,'
U
!
I
i
'--
\0
"0:)
100
H
c,')
ff)
..
.
....
(/)
~
(")
-------'-'- -j----.-
H
[-I
;.J
1':1
H
~~
EI
CJ.;
,-.- ·.
~
"I':-.
--_.~-
r·-·-----~-----
i
I
H
I
C:'
I
~
~
I
2i
H
r:..::
,0..~
.
~
E-l
,.,.,
OJ
"
~~
P".
..
r-.,
II")
Ci
W
H
~:
H
e
..--l
<:>
lf) .
U
1-'"1
r·,
c,')
~
'-'
C'l
;:J.
("')
(/)
~w.!
"-i~
r:.J
L~
(j
H
Jlr.l
,....
"d
Q
~
I1l
'-'
<J
r-,
~:.
.
Lf"I
."
..------
M
.... ..
..--l
~
--r._." . .
0
lit
<l
p..,
_-~-
-.
..;
~
, ....
~
l
J
~!
.--1
;:J.
(')
I
~
til
r::l
~
.~
~-1
H
tr1
~'1
, <.;
~1
,"1
,'E",
~.
tn
G
C"
C';
~
(')
L'1
'c'
o •
<'J
r,
t.:J
0
E- :
(/)
H
~
-:
~
--~--_._._-----I-~._._---
j
.,
,<J
C'
~
.. ---
.."
101
give a larger (smaller) estimate of
6
than the other
estimators.
(iii)
Assuming normality of the mean of the 200 sample estimates
~,
6, with an estimated standard error (SE) given by
=
SE(6)
SE(~)/1200,
when the PHF model is valid (Table
4.1c and Figure 4.3) there is no evidence that the bias in
6
is significantly different from zero, for all the cases
investigated.
In contrast to the conclusions suggested by the results of
Section 2.5.1,
we find that
mean square, and that
6
6 , not
2
6 , has the smallest
3
does not always have the smallest
1
bias.
(iv)
A comparison among the absolute values of the deviations
defined in Section 4.1, suggests that
A2[~(t)],
(a)
the unweighted average of the
closely compares to
A3[~(t)],
(b)
Only'for
6
closer to
4.2.3.
2
and to
6
4
than
IS,
most
6(t )'s, most
i
6 ,
1
A14[~(t)]
is the expected average
4
1
6 ,
3
the weighted average of the
closely compares to
(c)
6
6(t.)
A~[~(t)], ~
= 2,3.
SSII Simulation Study.
The simulation study for Sample Scheme II is much simpler to
design.
death
g
For each of
Tgj
= 1,Z,
j
gl
individuals from
TI ,
g
we generate a time of
from the distribution with hazard function
l, ... ,r
death
death from
r
and
TIl
and
gl
{t
.
Zj
~g(t;~gl'~gZ)'
We then order the resulting observed times of
}' giving
TI '
Z
t, the vector of (ordered) times of
We can then determine the corresponding
102
vector
p.
The numbers at risk of dying at each of the
k
=
r
+ r 2l
ll
times of death are given by
= rn -
L P.
(4.2.7)
j <i J
and
r
2i
Estimation of
= r 2l -
L (1
j <i
'" ,
p.)
J
(4.2.8)
1, ... ,k.
i
and calculation of the "averages" of the delta
6.
function at the times of death are as described in Section 4.2.1.
These steps were repeated for ZOO samples.
4.Z.4.
Discussion of SSII Simulation Study.
A summary of the first two sample moments of the four estimators,
for
(or
metric forms of
k
= 40), and for three choices of the para-
~g(t;t,;gl,t,;g2)
is given in Table 4.2.
respective delta functions are given in Figures 4.4-4.6.
Plots of the
Included in
the figures are histograms representing the 200 sample estimates of
6.
given by the four estimators; their sample means are also indicated.
From these results we may reasonably conclude the following.
(i)
The estimators
6. ' 6.
Z
3
and
6.
4
are similar in regard
to their first two sample moments.
6.
(ii)
6.
3
1
and
6.
4
The distributions of
are also very similar.
usually has the largest variance, but for an example
not reported (the delta function of Figure 4.4 and
r
(iii)
ll
= r ZI = 10) the variance of
6.
2
was largest.
The variances of the estimators decrease as
and
increase in the same ratio.
(iv)
Assuming normality of the mean of the ZOO sample estimates
103
TABLE 4.2
SUMMARY OF 200 SAMPLE ESTIMATES OF "6" UNDER SOME
SPECIFIED PARAMETRIC MODELS, SSII
Model specified by two Weibu11 distributions,
(a)
lJ
6
1
1
(t;50,1.5)
Mean
SD
t,
6
and
lJ
2
(t;60,2.0), r
ll
= r 21 = 20
{A2[~(t)]}
d2(~'~)
d3(~'£)
(.5466)
(.5961)
A
d 1 (~,£)
A1[~(t)]
{A3[~(t)]}
.5926
.0908
.5963
+.0037
-.0460
.0035
.5560
.0855
.5853
+.0293
-.0094
.0400
.5542
.0837
.5567
+ .0015
-.0075
.0419
.5543
.0839
.5415
-.0128
-.0076
.0418
~
6
2
~
6
3
-
6
4
Model specified by two Gompertz distributions
(b)
lJ
6
1
1
(t; .005, .05)
and
lJ
2
(t ; .008, .04) , r 11 = r
21
20
(.4791)
(.4520)
.4547
.0999
.4520
-.0027
.0245
-.0026
.4703
.0925
.4489
-.0214
.0088
-.0183
.4714
.0906
.4730
.0016
.0077
-.0193
.4713
.0907
.4693
-.0020
.0078
-.0193
~
6
2
~
6
-
6
3
4
(c)
PHF model specified by two Weibu11 distributions
lJ
6
1
1
(t;50,1.5)
and
lJ
2
(t ; 50,1. 5) , r
11
= r 21 = 20
(.5000)
(.5000)
.5140
.0818
.5000
-.0140
-.0140
-.0140
.5116
.0770
.5000
-.0116
-.0116
-.0116
.5112
.0763
.5000
-.0112
-.0112
-.0112
.5112
.0764
.5000
-.0112
-.0112
-.0112
~
6
2
~
6
3
-
6
4
104
H
H
U)
U) ,
U)
~
0
H
~
~q
'·-·-··----·--··;--r'····'- --
~
i
H
I
(I)
H
l=l
t::l
:j
H
(
~
51
Q
~
0'
,.J
:>-<
~
Q
N
...
.
!
,
"~I·
'"
,/
--:---[3-----'-r
./...
0'
I
;
,
:
______.._.i
J
I
H
U
i
1:'.
U)
..;:t
U)
~3
,.....
H
§
0
H
~
.u
<J
'.
,-.
,..=-i
.-I
~~
J=":
.:.
("')
<l
~
~
:,
.-I
I
~...
f'-,:
;:l.
~
",
"----------;;.--. r-----"
-._-
.'----~----------~'-1 ~
Lf"I
",
~:
0
.,Jf-t
s
U)
.
~
"~-t
f'
H
[ ..
U)
~1
~
eJ
~
rlJ
0
0
N
~""l
0
(Il
~
C"..~
0
H
U)
H
~
1,
,
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,,
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I
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H
~
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....:l
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FIGW£ 4.5
ESTI?~TI:S
HISTOGRAHS OF 2DO SR·U'LE
OF
fl
(e)
1
t:,
..13K'!
(t;.005,.05)
fl
Z
I'
I
rl'
r··
"(t)
.\
I ' ' ·..
•
uun_n
I
1
_, •••• ,
to.
S"'~."L:: ~.\:.
t
~i
.,
:,;
r
i·'·
1
!
~I
i
'
~r
-,'
LJ
.1
i
I
!
~:':l
"
..
,j)
....,
(c)
~--------r-I
~~
"
L".
i
,
I
I
!
,i
I'
:
1.•
,.n
i--.-r-'
........ " ,
~ f'~~---""---u,,~
,._, , , ,.,
tln".
I
'
\CI
1
"n
J
I
L
~2
I
t·'
I
SSII
-
!
I
.i
L]
GO~ERTZ DIST!UBUTIO~S,
(t;.!)08,.04)
(b)
;:
.
and
~:
J
~
,
t,(t) IS Sl'ECIFIF2) BY r!Q
1..
!
!
r-'
....-J.
I
i
:." '." -"
L..,
'----,~yr
i~
t,
i! .•
I
L
I
I
I
i
::t'f
"
----'
~
I
I
_ _- . I
-.
L •
~
v----~
~
I
:
'
,..4
<::\'7,-::-,:.'::
.
i
I
I'
L ,
-i"
., I
I
..
~
i
"
.,
TI.....-:
!J
~
:,
.....
o
VI
106
H
H
UJ
UJ
.
tf.l
:-:-:
'."-'" '-r-"'---
C"
H
t~
~
·..
'!
H
-
p~
H
:1
j
.. j
,,
H
P
H
t~.
"
~
l:
U)
·-1·-+··-~·
'I
.,
"
"
!
"
~
~
H
~
.-2
~
, .J
£:·1
~I
........
~
rl
~
H
...
l'")
(')
J:Ll
L')
H
U
+J
~
'--'
trJ
;:I.
,,'
tf.l
H
'"0
~J
........
p~
",.
0.
~
C;
H
J:Ll
+J
'-"
N
g
........
I,..,
.-,.,, , ..
<l
r-I
~:1
..:..
t~
0
<l
........
~
~
.
._......,. - . _ - - .
--
~
.
--.--~
---
~
~
l!"l
~j
~j
+J
r",
'- ·'--1.-. _-
r-~
;:I.
0
"
U)
r··,
H
:,.J1
.r-~
H
tf.l
4l
r'j
,.
8
~
u~
C':;>
0
N
~L<
("'
UI
~,
~
~~
(,)
0
H
tf.l
H
...
,'T'
, -1
_. -
107
~,
~,
when the PHF model is valid there is evidence that the
~1.
bias is significantly different from zero for
~1
however, that
is in fact, unbiased.
We know,
The conflicting
conclusion must, I think, be attributed to sampling variability for this particular parametric model and numbers
initially at risk of dying in
(v)
n
1
and
n .
2
A comparison among the deviations defined in Section 4.1
suggests that
A2[~(t)]
(a)
most closely compares to
but in some cases
4.2.5.
is closer to
~4'
and
~2
for
= 3,4.
t
~l
(b)
Alt[~(t)]
~2' ~3
most closely approximates to
Some Measures of the Difference Between the
and an Estimate
~
A [0(t)].
3
~(t)
Function Model
Based on the PHF Model.
There are many ways in which we can define a measure of the difference between the delta function and an estimated constant
on the PHF model.
t, [T ,T ], say, are of form
1 2
~)dt,
d
~(.)
(4.2.9)
is a well behaved continuous increasing function of the
I~(t) - ~l.
absolute difference
over a restricted range of
zero or one at
=0
t
or
tion of the integral of
some limit as
based
Simple measures of this difference over some
restricted range of
~lere
~
t
+
00
t
t
It is necessary to consider
~(.)
because the delta function is frequently
=
~(.).
00,
and may greatly influence the evalua-
Also, the delta function may approach
and we may have little interest in the difference
108
~(t)
between
stant.
~
and
t
t
~
00,
Restricting the range of
function, while taking
as
as
T
2
=
00
which itself would approach some cont
would ignore this part of the
\fI(.)
would simply give us the limiting value
~ 00.
Two possible choices for -the form of
\fI(.)
are
IMt) - ~I
. (i)
(4.2.10)
and
(ii)
(4.2.11)
A serious criticism of any of these measures, however, is the
arbitrariness of the
and
values.
Some preliminary results
from the Monte Carlo Studies described in Sections 4.2.1 and 4.2.3 in
which
was defined by (4.2.11) indicate that the measure
\fI(.)
very dependent upon the values for
T
I
and
T .
2
d
is
Until further investi-
gations are made we do not recommend using measures of this type.
4.3.
Alternatives to the PHF Model.
In Chapter II we considered estimation of the parameter in the
PHF model, but in Chapter III we found that in many situations the HRF
was not constant.
mators of
~
In Section 4.2 we examined properties of the esti-
(derived under the PHF model) when the HRF was not con-
stant.
On the basis of the results presented in Chapter III we will
suggest the following procedures as practical methods for comparing
survivorship functions.
These suggestions follow two lines of thought.
They are given as a possible basis for practical application and a topic
for research and development.
109
4.3.1.
Alternate Models.
We may propose some simple models as alternatives to the PHF
model.
We can then construct tests of the validity of the PHF model
with regard to these simple alternatives.
For example, suppose that the hazards in two populations are not.
proportional, but have a linear additive relationship such as
e
>
(4.3.1)
0,
or more generally,
(4.3.2)
\,11 (t)
If either (4.3.1) or (4.3.2) were valid we see that the HRF would be
given by
e (t)
(4.3.3)
or
(t)
lJ
1
-e (t) = lJ (t)
2
e+
<P(t)
lJ
2
(t)
(4.3.4)
,
respectively, both of which are functions of time.
Notice that the
HRF given by (4.3.3) is a monotonic decreasing function if
a monotonic increasing function, with a limiting value of
lJ
e.
2
(t)
is
From
the analytic results of some HRF's summarized in Table 3.5, this behavior of the HRF (or delta function) is not unreasonable for some pairs
of parametric models (e.g., Makeham Gompertz, linear hazard distribution, truncated logistic, gamma, and Burr).
no
Let us assume the linear additive model given by (4.3.1) is
valid and construct a test for the appropriateness of the PHF model.
H : 1> = 0
O
More specifically, we wish to test
H : 1>
~
A
O.
against the alternatives
We will briefly describe two tests of this hypothesis
which are not based on any a priori knowledge of the forms of the
separate hazard functions.
Recall from Section 3.2 that we can obtain estimates of the hazard
function by several methods.
One such nonparametric method for grouped
data, using an exponential approximation to the survival function within
intervals for
is
n ,
g
1
"h. R-npgi'
(4.3.5)
1
where
p . is an estimate of the conditional probability of surviving the
gl
t., g
interval [ti,t _ ), given alive at
i l
interval is
t~.
1
=
1
1,2.
The midpoint of the
We may then substitute these estimates into (4.3.1)
and use a weighted least squares method to estimate
e
¢.
and
The
weights usually chosen are the reciprocals of the variances of the
corresponding error
t~rms
in the model.
estimate of the approximate variances of
e
and
¢
WI (t~)
[WI (t~) -
are independent), the variance of
unknown parameters
Although we can obtain an
eW2(t~)
(which
- ¢]
involves the
which we wish to estimate.
Use of these
weights in an iterative procedure is a natural way to proceed.
While this method may prove enlightening with regard to a test of
H '
O
we are overlooking the fact that the estimates
W2(t~)
are both random variables.
estimation of the parameters
error in testing
H .
O
e
and
Ul(t~)
and
This may not be very important in
¢, but could be a source of serious
Some of the techniques
develo~ed
in structural
111
regression (Kendall and Stuart, Vol. II, Chapter 29, 1972) could possibly
be adapted for this problem.
In particular, we may have to make some
assumptions regarding the variances of the observed
~ (t'.),
g
1
g
1,2,
before we can obtain estimates of the parameters in the model.
4.3.2.
Subdivision of the Follow-up Period.
Without going even as far as to specify an alternative to the PHF
model we can suggest an "indirect" test for the appropriateness of the
PHF model.
Sometimes it is possible to divide the follow-up period into
a relatively few subperiods within which a PHF model may be appropriate.
If this is possible the estimates of
periods can be compared.
6.
(or
e) for successive sub-
For a sufficiently large sample size, we may
assume that the estimators
6.
(or
e) have a normal distribution.
(approximate) estimate of the variance of
6.
(or
An
e) can be derived
using an approximation to the variance of the ratio of two random
variables (e.. g. see (2.2.2)).
equality of the values of
estimates so obtained.
6.
We may then formally test for the
(or
e) over successive periods, using the
CHAPTER V
SOME APPLICATIONS OF THE PHF MODEL
The purpose of this chapter is to illustrate the techniques of
estimating the parameter
example.
/),
in the PHF model, using an empirical
We should be aware that this is not a complete analysis of the
data, but only one aspect of the analysis.
The estimation of
/),
in the PHF model, ].11 (t) = 8].12(t), where
/), = 8/(1 + 8), is straightforward.
It requires a reasonably simple
computing algorithm to obtain the vectors of values
~hen
and
p
we are given the length of time of follow-up, population identifi-
cation, and possibly indication of censoring, for each individual under
observation.
Before proceeding with the example we will give an approximation
to the variances of the estimators of
/),
derived in Section 2.4 which
allow for ties.
5.1.
Approximations
~
the Variances
~
the Estimators of
6
Which
Allow for Ties.
For convenience, throughout this chapter we will again use
to denote the ML estimator
/),4-
/),4'
An approximation to the variances of the estimators of
tional on the numbers at risk in
6, condi-
is given by (2.Z.Z), i.e.,
and
k
2
. I 1 anorlorZO
~=
L~
~
~
(5.1.1)
113
where
~l
k
=
I
a ·r · p ·,
i=l l 1 21 1
set of weights corresponding to
6 , l
l
= 1, ... ,4.
We now consider an approximation to the variance of the estimator
of
6, 6'
say, which allows for ties.
Recall from Section 2.4 we
derived the estimator
6'
where
d.
(5.1.Z)
is the number of deaths from
gl
1T
occurring at
g
"t
i
It
'
k
g = 1,2, i
= l, ... ,k, {ail
I
is a set of arbitrary weights, and
L·
i
i=l
Conditional on
and
The distribution of
d
1i
r , the random
2
(or
d
2i
varia~les
are
), conditional on
the total number of deaths observed at
"t"
i
'
and
ali
d
i
= d
+ d 2i ,
1i
is difficult to determine.
A very rough approximation to this distribution is to assume
d
r
binomially distributed, i.e., d Zi : b[di,qi1, where
i
d 2i ·
qi
Zi
is
Zi
1, ... ,k.
Let
(5.1.3)
y
(5.1.4)
(5.1.5)
It can be shown (see Appendix C), using the assumption
d Zi
is bino-
mially distributed, that
E[X]
-
Ia~ [d.r '
.11
1
Z1
+ (1 -
r
2
' - 1
di)d.q. + d.(d i - l)q.],
1
1
1
Z1
(5.1.6)
114
and
\a~[(l +
E[Y] ,;
r
. - d.)diq.
1
i l l
1
+ d.(d. - l)q:"I.
111
(5.1.7)
1
Also
Var[X] ,;
La~Z([l
-
+
r
+
(r
d.)]Zd.q.
i lZil
+
4(Zd
i
1
- 3)d (d i
i
Zi
1
l)q~
+ 2(3 - Zdi)di(d i -
l)q~.
(5.1.8)
Var[Y] ,;
La~Z([l
. 1
1
+
[6d~ ~
+ r 1. - d.]Zd.q.
111
1
7
+
(r
1i
- d.)(5d. - 6 - r
1
1
Z
.)]d.q.
11 1 1
+ 4(Zd.1 - r 1: - 3)d.(d.
- l)q~ + 2(3 - 2d.)d.(d. - l)q;).
111
111
~
1
(5.1.9)
and
~z( [1
Cov[X,Y] ,; Ia
• 1
1
+{6d~
~
+
+ r 1. - d.][l
- (r ' + d. )]d.q.
1
Z1
111
1
' + d.) - (r . - 7 + (3 - Zd.)[(r
11
11
Z 1
Z(3 - Zd.)d.(d. 1
-1
1
l)q~).
1
An approximation to the variance of
Var
[~ ']
== {
d.)]}d.q~
111
E [X] } Z{Vad X ]
E[X + Y]
{E[X]}Z
(5.1.10)
-
~'
is
+ Var[X + Y]
{E[X + Y]}Z
Z Cov[X,X + YJI
E[X]E[X + Y]
f'
(5.1.11)
115
Substituting (5.1.6)-{5.1.10) into (5.1.11), where
=
EIX + Y]
= VarIX]
VarIX + Y}
E[X] + E[Y],
+ Var[Y] + 2 Cov[X,Y],
and
Cov[X,X + Y]
= VarIX}
+ Cov[X,Y],
gives an expression for the approximate variance of
true
6
(or
to be
Since the
is unknown, we take a further approximation and sub-
e)
stitute the estimate of
{ail
6'.
6
in (5.1.11).
We will take the weights
{a ti }, the weights corresponding to the estimators of
with single times of death, t
6
1, ... ,4.
I t should be noted that the approximation given by (5.1.1) for
~
Var[6 }
4
is equal to the asymptotic estimators given by the expected
value of the negative reciprocal of the second derivatives of the loglikelihood function {see (2.2.11», under this assumption about
~1en
d
2i
.
we have censored observations, we make the assumption that
the effective numbers at risk of dying at each of the times of death are
r' .
gl.
where
and
using
c
gi
t-i+1'
r' .
gl.
r gl.. - ~c g i'
g
= 1,2,
is the number of individuals from
Approximations to the estimators of
for
r
gi
l, ... ,k,
i
'IT
(5.1.12)
censored between
g
6
t.
l.
will be obtained
in the respective formulas, including the approxi-
mations to the variances of these estimators.
116
5.2.
Example of Mortality Data Among _Hourly Horkers in _the Rubber
Industry.
The data are from a retrospective study of hourly workers in the
rubber industry collected by the Occupational Health Studies Group
(OHSG) of the University of North Carolina.
A subset of the data file,
provided for the author by the OHSG, includes information on length of
follow-up, age and number of years of previous work experience in the
rubber industry at time of entry into the study, and survival status at
the end of the study, i.e., alive (including lost to follow-up and
censored) or dead.
The subset contained 3907 white male employees
between the ages 30 and 55 at entry into the study, from a major tire
manufacturing company in Akron, Ohio.
The study period was ten years
between January 1, 1964 and December 31, 1973.
in
a few
cases and ties may exist.
The data are incomplete
We will not distinguish between
censored observations resulting from termination of the study and
voluntary withdrawal,
Le.,· loss of an individual from
t~e
study.
It is of interest to ascertain whether exposure to the working
conditions in a tire manufacturing plant has a deleterious influence on
survival of the employees.
mortality.
Age is, of course, an important factor in
We conjecture that exposure (acquired during work experi-
ence) is also an important factor.
A description of one approach to comparing the survivorship
functions of men with different amounts of exposure, controlling for
the effect of age, is now given.
Define six populations as follows.
TIl:
men aged 35-40 with 5-15 years of exposure.
TI :
2
men aged 35-40 with 15-25 years of exposure.
117
11
3:
men aged 40-45 with 10-20 years of exposure.
:
men aged 40-45 with 20-30 years of exposure
:
men aged 45-50 with 15-25 years of exposure.
11 6 :
men aged 45-50 with 25-35 years of exposure.
11
11
4
5
One way of making this comparison is to compare the survivorship
functions of
with
11 2'
11
3
wi th
4' and
11
with
We can do
derived under the PHF assump-
~
this using any of the estimators of
TI 5
tion.
A secrond way emerges if we think of
five further years have elapsed and
TI
5
11
3
as being like
as being like
11
1
TIl
after
after ten
years have elapsed, i.e., the cohort of men 35-40 years of age with 5-15
years of work experience will be age 40-45 with 10-20 years of experience five years later and will be age 45-50 with 15-25 years experience
ten years later.
We may think of
11
4
and
11
6
with regard .to
11
2
in
a similar way.
The estimators of
6
obtained in the three comparisons above
provide, in a sense, information on twenty years of survivorship for
the two populations of men 35-40 years of age with 5-15 and 15-25 years
of exposure.
This approach assumes that hazard rates observed for the
different
groups between 1964 and 1973 will be those in the next ten
ag~
year period.
It also implies all other factors relating to calendar
time, e.g., environmental and health conditions within the plant and
associated community, are the same.
These assumptions are not likely
to be true, but this may have little effect on the comparisons suggested.
118
A summary of the data for each of the six populations described
above is given in Table 5.1; Table 5.2 gives a summary of the estimates
of
~
for the three proposed comparisons of populations.
TABLE 5.1
SUMMARY OF DATA ON SIX POPULATIONS OF WORKERS IN A
MAJOR TIRE MANUFACTURING COMPANY, AKRON, OHIO, 1964-1973
Population
Number
Alive
at to
Number of
Deaths
During
Follow-up
TIl
295
9
.0305
67
.2271
TI
541
14
.0259
85
.1571
458
21
.0459
58
.1266
458
22
.0480
50
.1091
681
50
.0734
75
.·1101
76
10
.1316
8
.1053
TI
TT
2
3
4
TIS
TI
6
Proportion of
Pop. at to
Dying During
Follow-up
Number of
Persons
Censored
Before
Close of
Study
Proportion
of Pop. at
to Censored
Before
Close of
Study
119
TABLE 5.2
ESTIMATES OF 6 FOR THREE
CO~WARISONS
OF POPULATIONS OF WORKERS IN A
MAJOR TIRE MANUFACTURING COMPANY, AKRON, OHIO, 1964-1973
6
Comparison
6
Populations
SE(6 )
1
SE(L't. )
2
SE(6 )
3
SE(6 )
4
.. 5509
.5481
.5482
.5481
.1065
.1059
.1059
.1059
.5000
.4905
.4905
.4905
.0766
.0763
.0763
.0763
.3504
.3490
.3491
.3491
.0794
.0787
.0788
.0788
C :
1
C :
2
C :
3
lf
lf
If
1
3
5
vs.
vs.
vs.
lf
lf
lf
2
4
6
1
6
L\ 3
2
4
We notice that there are no practical differences among the three
estimators
6 , 6
3
2
and
6
4
in terms of their estimate of
standard error of the estimate.
Tests of the hypothesis
6
HO: 6
and
= .5,
which implies there is no difference in the mortality of the two populations, were performed assuming normality of
6~,
~
= 1, ... ,4, with
an approximate estimate of the standard error obtained from (5.1.11).
None of the tests for the three comparisons was significant.
We suggest that the conclusions of no differences among the
mortality of the populations be accepted with caution for the following reasons.
The estimates of
differ from the estimates of
6
6
for the third comparison seem to
for the first
t~o
comparisons.
We
do not know whether this is a real difference which our test procedure
failed to detect or whether it reflects the influence of (i) unequal
120
sample sizes at
to' (ii) our method of adjusting for censored observa-
tions, particularly at the last time of death, or (iii) the elimination
of men who retired early, most of whom would have been in
and
W
s
w .
6
We are uncertain what influence censoring may have on our estimation.
In this example most of the censoring occurred at the close of
the study, i.e., at the same fixed point, and we adjusted the risk sets
at the last time of death to account for this.
In an estimation not
presented, we omitted the last time of death from our calculations.
There is some indication that this affects the estimates of
standard error of the estimates only in a small way.
6
and the
This point needs
more investigation, however, before we can make any definitive statements regarding the influence of censoring on our estimation procedure.
We conclude by suggesting that these results be taken as provisional and proceed with additional methods of analysis.
we may take the
r~sults
analysis similar to that
In particular,
and use them to motivate a regression type
di~cussed
by Cox (1972), or possibly some
parametric model which allows for covariables.
CHAPTER VI
SUGGESTIONS FOR FURTHER RESEARCH
We have indicated areas of further research in the comparison of
survivorship functions at various points throughout this dissertation.
We briefly described (Section 4.3) some methods for detecting departures from the PHF model in Chapter IV.
These were regression tech-
niques, and, in particular, methods of structural regression.
We
pointed out that the typical weighted least squares approach to estimating parameters in an alternate linear additive model has two main
limitations.
One is the choice of weights since an approximation to the
variances of
~2(t)
meters in the model.
over appropriate intervals depends on the paraIn general, moderate errors in weights have
relatively little influence and fair approximations can be used with
some confidence.
When the weights depend on the parameters to be
measured, natural iterative methods should be considered.
A second
limitation is that the regressor variables, in addition to the response
variables, are random variables.
Some of the techniques developed in
structural regression could.possibly be adapted for this problem.
However, application of structural regression requires introduction
of reasonable assumptions on the relative magnitude of errors in the
variables concerned.
This needs detailed study.
The construction and analysis of models alternate to the PHF
model appears to be an area offering considerable possibilities of
122
research.
Suitable models should become apparent after sufficiently
prolonged study along the lines described in the previous paragraph.
Possibly preceding this there may be scope for detailed theoretical
analysis of certain simple models similar to the models described in
Section 4.3.1.
Indices of the kind described in Sections 4.1 and 4.2.5 (representing variation in the HRF or some function of the HRF) might prove
useful in deciding on suitable intervals of time within which a PHF
model might be regarded as a sufficiently good approximation to use.
One might, for example, have a requirement that for this to be so, the
"index" for any such interval should not exceed a specified value.
Determination of such a critical value would of course call for a
careful assessment of the consequences of its use.
We have suggested a very rough way of allowing for censored data
(Section 2.4.1).
Some of the calculations in Chapter V indicate that
assumptions on the effect of censoring may, in some circumstances,
have a noticeable effect on the results obtained.
A thorough inve8ti-
gation of these effects seems likely to produce interesting contributions.
Of course, even with such studies, it is by no means certain, or
even likely, that a universally "best" (or even good) way of allowing
for censoring will exist.
When we really do have full information on
how censoring operates in a particular case we can try to make as
appropriate an allowance for it as we can.
Very often, however, we can
do no more than make a more or less enlightened guess.
BIBLIOGRAPHY
Abramowitz, M. and.Stegun, I. A. (eds.) (1972), H~~dbook of Mathematical
Functions' with Formulas, Graphs and Mathematical Tables, National
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APPENDIX A
MOMENTS ABOUT ZERO OF SOME LIFETIME DISTRIBUTIONS
Let
T
be a positive continuous random variable with cumulative
distribution function
fT(t)
d
= dtFT(t)
FT(t)
exists.
such that the probability density function
It is known that the mean of
T
is the inte-
gral of the survival function, i.e.
f
E[T]
where
PT(t)
=
1 - FT(t)
00
o
00
[1 - FT(t)]dt
=
f
PT(t)dt,
(A.l)
0
is the survival function.
To show (A.1) is true, by definition we have
E[T]
= -f
00
o
d
t--[l - FT(t)]dt.
dt
Integration by parts of (A.2) leads to (A.1) if
(A.2)
lim t[l - FT(t)]
= O.
t-l<Xl
I t can b e s h own t h at t h e
'
y th moment a b out zero can b
e wrltten
as
(A.3)
u'
y
provided
A1.
Makeham Gompertz Distribution,
For a derivation of the first two moments about zero of the
Gompertz distribution
(A
0)
see Hoe1 (1972); the mean of the Gompertz
distribution has been given by Broadbent (1958) and by F1ehinger and
Lewis (1959).
The moments of the Makeham Gompertz distribution are not
readily found in the statistical literature and are therefore presented
130
here.
Using (A.l) the mean is
E[T]
=f
u'
1
OO
B
{exp[-(-)(e
Kt
K
0
- 1) - At]}dt
To evaluate (Al.l) consider the transformation
u'
1
B K
e / foo e -(B/K)Y
= ---1
K
y
Y= e
Kt
(Al.1)
which gives
-(l+A/K)d y.
(A1.2)
The exponential integral (Al.2) may be written in terms of an incomplete
gamma function, giving
u'
1
(Al. 3)
00
where
r(a,x)
J e-tta-Idt
is the form of the incomplete gamma func-
x
tion given by Abramowitz and Stegun (1972) (p. 260, [6.5.3]).
For higher moments we can show that (A.3) is equivalent to
u' = y(-l)Y
y
y
-1 d
-l
,
ul
----"
dAy-l
CAL 4)
The variance is then
Var[T]
wh~re
u'
2
B K
= 2e 2/ fooe -(B/K)Y(o~ny ) Y-(l+A/K)d"y.
K
(Al. 5)
1
Evaluation of both (Al.2) and (Al.5) requires numerical methods.
A2.
Linear Hazard Function Distribution.
For a derivation of the first two moments of the linear hazard
function distribution see Kod1in (1967).
The expressions given below
are equivalent to those given by Kodlin, but are somewhat simpler in
131
form and may be evaluated using techniques for the standard normal
integral.
~sing
(A.l) the mean is
00
E[T]'= u' =
1
J {exp[-at
0
2
- ~8t ]}dt.
(A2.l)
To evaluate the integral in (A2.l), complete the square of the exponential factor and let
y = tlB+ a/lB.
We find
(A2.2)
(A2.3)
where
¢(z)
is the standard normal probability integral.
For higher moments we can show that (A.3) is equivalent to
E[TY]
=
u'
Y
= '1(-1)
d'1 - l u ,
l
y-l
da'1-
1
(A2.4)
The second moment about zero is
U
2 ~ - ~a2/2B[ J¥J{l
i
= h -
- $[~]} + 2ea2/2B[ Jr]~[~]~
aui L
(A2.5)
The variance is
Var[T]
A3.
Truncated
(A2.6)
Logisti~ Distributio~.
The mom.ents of the truncated logistic distribution do not seem to
be published.
Consider the standard logistic density function iruncated at
Y
K,
132
p(y) = e- Y[l
F(K) = [1 + e
where
-K -1
]
.
+ e- Y]
-2
1[1 - F(K)],
(1\3.1)
The moments about zero are given by
(A3.2)
-K -1
where
C=[l-(l+e)
],
Using the relationship given by
00
I
(_l)j-lje- jy
(A3.3)
j=l
(see Johnson and Kotz, Chapter 22, p. 4), we can write (1\3.2), after
interchanging the integral and summation, as
(A3.4)
If
Y is an integer we can evaluate (A3.4) by putting
z = jy
and then
integrating by parts, leading to
(A3.5)
where
Sy(Kj) = e-Kj{l +
(~1)
(K1~2
+
+ ... +
.
i~Y}
(A3.6)
J
For
(Kj) > 0
we can evaluate
S (Kj)
. Y
using the table of Poisson
probabilities.
If we want the distribution of
that
unless
T
is truncated at zero, then
E;,
< 0
probabilities.
we cannot evaluate
T
Y is truncated at
S (Kj)
Y
=
E;,
+ BY
K
=
such
-E;,IB.
using tables of Poisson
Thus,
APPENDIX B
BEHAVIOR OF THE
HRF
FOR PAIRS OF DISTRIBUTIONS
FROM THE SAME PARAMETRIC MODEL
BI.
Makeham Gompertz.
The HRF, using the notation in Table 3.4, is
KIt
e(t)
=
Al + B1e
(Bl. 1)
AZ + BZe
Let
t
e.
y
Assume
G(y)
root.
Then
0 < K < K
I
2
= d~;Y)
Now, G(y)
and consider the behavior of the function
is a decreasing function and has at most one
There is a root if
G(l)
since
Kzt
lim G(y)
{AZB1K1 - [A 1 BI K2 + (K 2 - K1)B1B Z]} > 0,
_00
.
If
G(y)
has a root, then we know there is some
y~
value of
y
(or
t), '1
say, such "that
G(y) > 0
for
y > '1
and
134
G(y) < 0
for
y <
n.
Thus, if
(B1,3)
for
0 < Kl < K2
B2.
Truncated Logistic.
we conclude that
e(t)
has a mode.
The HRF, using the notation of Table 3.4, is
e(t)
(B2.l)
For convenience, we will consider
log O(t).
We have
d log e(t)
dt
(B2.2)
We see that
d log e(t) > 0
dt
«
0)
Bl
G(t) < -B2
if
Bl
(G(t) > - )
B2
[1 + exp -B~- ]
[1 + exp~]
l~-l:JJ
t
-
where
E, ]
[
G(t)
To simplify, let
Then
H(y)
K
(B2.3)
81
= -- and
82
y
=
Choose
81 > 82 > 0
and consider the behavior of
H(y).
We have
(B2.4)
135
Now, hey)
clearly has a root, since for
function of
y, h(O) = -a
some
y < n, say, and
Le.
81
H(y) < 82
d log e(t)
> 0
dt
in general, for
For
If
81
for
«
= 82'
or
for some
2
K
= 1,
B3.
t < n
for
°
for
we see that
81
a
< a
l
«
Z
H(y)
(a
l
has a mode,
Hence,
(t > n *).
= -Q,nn
e(t)
H(y)
Y > n·
1
1c
we see that
d log e(t) > 0
dt
e(t)
That is, hey) <
Thus
y > n·
And in
always is unimodal.
however, we can show that
(decreasing) function if
Thus
for
81
H(y) > -82
y < nand
8i I 8
lim hey) = 00,
is an increasing
y-¥XJ
h (y) > 0
0)
81 = 82 ,
and
l
K > 1, hey)
e(t)
is always monotonic.
is a monotonic increasing
> a )' or that
Z
0)
is a monotonic increasing (decreasing) function for
t,;l > t,;Z
Burr.
The HRF, using the notation of Table 3.4, is
-a
alBIC} +
t
Z]
a ,8
e(t)
g
g
> 0, g
1,2.
(B3.1)
Then
d8(tl
dt
Let
h(t)
h(t) > 0
lim h(t)
t-¥XJ
(B3.2)
Then
« 0).
00
We see that
(- 00)
if
h(O)
= al
~(t) >
cit
°
«
0)
as
- a ' h(t)
Z
Suppose
a
1
> a ; we also
Z
136
:)bserve that
t
as
> 1
or
t
< n
t
a. -1
dh (t)
dt
< 1.
(t > n)
ment is similar for
-
t
1
.]
So there is some t, n
for
Thus
is positive or negative as
say, such that
8(t)
d8(t) > 0
dt
is unimodal.
«
The argu-
0)
APPENDIX C
APPROXIMATE VARIANCES OF THE ESTIMATORS OF
Consider the estimators of
~
WITH TIES AND CENSORING
~
given in Section 2.4 which includes
tied observations, i.e.,
~I
with
I
i
(C .1)
k
L
i=1
Assume
d 2i
b[di,qi]' where
di
d
li
+ d 2i ,
1, ... ,k.
Let
(C.2)
and
An approximation to the variance of
Var[~I]
where
~I
is
_ 2 Cov[X,X + Y]l
E[X]E[X + -yf-J'
{E[X + y]}2
(C.4)
= [ E[X] 12[var[x] + Var[X + Y]
E[X + Y]J
{E[X]}2
E[X + Y]
Var[X + Y]
= E[X] + E[Y],
Var[X] + Var[Y] + 2 Cov[X,Y],
138
Cov[X,X + Y] = Var[X] + Cov[X,Y].
'lnd
Then
E[X] = Ia~E[u.],
E[Y] = Ia'.E[v.],
.11
.11
1
1
Var[X] = La~2Var[u.],
Var[Y]
.11
1
= La~2var[v.],
. 1
1.
1
and
Cov[X,Y]
= La~2cov[u.,v.],
. 1
1
1
1
where
u
i
=
(d i - dZi)(r Zi - d Zi )
= d i r 2i
Z
- d Zi (r
+ di) + d Zi '
2i
(C.S)
and
(C.6)
\ve will drop the subscript
of
u
and
v
i
and proceed to find the first two moments
and their covariance.
The following relationships between
moments about zero and factorial moments of the binomial distribution
employed in the derivations are given below.
where
E[d~]
E[d(2)] +E[d ]'
2
Z
E[d;]
E[d (3)] + 3E[d(2)] +E[d ]'
2
2
Z
E[di]
E[d(4)] +6E[d(3)] + 7E[d(Z)] + E[d ]'
2
2
Z
z
E[d(Y)] = d(d - 1) .•. (d - Y + l)qY.
Z
Thus we find
E[u]- dr
dr
2
2
(r
2
+ d)dq + d(d
+ [1 - (r
Z
l)q
Z
+ c1q,
+ d)]dq + d(d - l)q 2 .
(C.7)
139
E[v]
=
(r
l
- d)dq + d(d - l)q
2
7 dq,
= [1 + r l - dldq + d(d - l)q Z .
(C.8)
We have
= d(d
- l)(d - Z)(d - 3)q4 + 3[Z - (r
Z
l)q
+ d)]d(d - l)(d - Z)q3
Z
+ [1 - (r
z+
2
d)] dq,
(C.9)
. and
{E[u']}Z
=
dZ(d _1)Zq 4 + Zd 2 (d - 1)[1 - (r
Z
+ d)]q3 + [1- (r + d)]ZdZqZ,
Z
(C.lO)
which gives
Var[u] = [Cd
Z)(d
3) - d(d - 1)]d(d
1)q4
+ {Cd - Z)[6 - 2(r Z + d)] - 2d[1 - (r Z + d)]}d(d - l)q 3
+ {Cd - 1)[7 - 6(r
+ [1 -
(r
Z
Z
Z Z
+ d) + (r Z + d) ] - d[1 - (r Z + d)] }dq
z
Z
+ d)] dq.
(C .11)
Combining like terms in (C.9) and (C.lO), upon simplification we have
Var[u] = Z(3 - Zd)d(d - 1)q4 + 4(r
+ [6d - 7 + (r
Z
Z
+ Zd - 3)d(d - 1)q3
+ d)(6 - Sd - rZ)]dq
(C.lZ)
Z
+ [1 - (r
Z
2
+ d)] dq.
140
We can obtain
Var[v)
and (Co6) that
u'
directly from (Co12) since we notice from (Co5)
and
v
are of the same form with
-r
replacing
li
Thus
Var[v)
2(3 - 2d)d(d - 1)q4 + 4(2d
r
l
- 3)d(d - 1)q
3
+ [6d - 7 + (r 1 - d)(5d - 6 - r )]dq 2 + [1 + r - d)dq.
1i
l
(Col3)
For the covariance, Cov[u,v)
=
Cov[u' ,v), we have
{4
3
2}
E[u' ,v) = E d 2 + [(r - d) - (r + d»)d - (r - d)(r + d)d
l
2
2
l
Z
Z
d(d - l)(d - 2)(d - 3)q 4
+ [6 + (r l - d) - (r + d»)d(d - l)(d - 2)q 3
2
+ [7 + 3(r l - d) - 3(r + d) - (r - d)(r + d»)d(d - l)q z
2
l
2
+ [1 + (r l - d) - (r + d) - (r - d)(r + d»)dq,
Z
l
2
(C.14)
and
E[u')E[v)
2
2 4
d (d - 1) q
2
+ {[l - (r Z + d») + [1 + r - d)}d (d - l)q
l
We then find, using (C.14) and (Co15), combining like terms, and
simplifying that
3
141
Cov[u,v]
= 2(3 - 2d)d(d - l)q 4
+ 2[2(d - 3) + (r
Z
+ d) - (r
1
- d)]d(d - l)q
3
+ {6d - 7 + (3 - Zd)[(r Z + d) - (r - d)]}dq
1
2
+ [1 - (r Z + d)][l + r 1 - d]dq.
(C.16)